A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a

intuitive definition of the limit

If all values of the function $f (x)$ approach the real number L as the values of $x (\ne a)$ approach a, $f (x)$ approaches L

A one-sided limit of a function is a limit taken from either the left or the right

A function has a vertical asymptote at $x = a$ if the limit as x approaches a from the right or left is infinite

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Calculus Volume 1 · 2.2 The Limit of a Function

The Limit of a Function: practice problems

31 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 31
For the following exercises, consider the function $f (x) = \frac{x ^{2} - 1}{∣ x - 1∣} .$
What do your results in the preceding exercise indicate about the two-sided limit $lim_{x \to 1} f (x)?$ Explain your response.
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Exercises 33
For the following exercises, consider the function $f (x) = (1 + x)^{1 / x} .$
What does the table of values in the preceding exercise indicate about the function $f (x) = (1 + x)^{1 / x} ?$
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Exercises 35
In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.
[T] $lim_{x \to 0} \frac{s i n 2 x}{x}; \pm 0.1, \pm 0.01, \pm 0.001, \pm .0001$
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Exercises 37
In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.
Use the preceding two exercises to conjecture (guess) the value of the following limit: $lim_{x \to 0} \frac{s i n a x}{x}$ for a, a positive real value.
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Exercises 39
[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.
$lim_{x \to 1} (1 - 2 x)$
x $1 - 2 x$ x $1 - 2 x$
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Exercises 41
[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.
$lim_{z \to 0} \frac{z - 1}{z ^{2} ( z + 3 )}$
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Exercises 43
[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.
$lim_{x \to 2} \frac{1 - \frac{2}{x}}{x ^{2} - 4}$
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Exercises 45
[T] In the following exercises, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?
$lim_{α \to 0^{+}} \frac{1}{α} cos (\frac{π}{α})$
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Exercises 47
In the following exercises, consider the graph of the function $y = f (x)$ shown here. Which of the statements about $y = f (x)$ are true and which are false? Explain why a statement is false.
$lim_{x \to - 2^{+}} f (x) = 3$
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Exercises 49
In the following exercises, consider the graph of the function $y = f (x)$ shown here. Which of the statements about $y = f (x)$ are true and which are false? Explain why a statement is false.
$lim_{x \to 6} f (x) = 5$
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Exercises 51
In the following exercises, use the following graph of the function $y = f (x)$ to find the values, if possible. Estimate when necessary.
$lim_{x \to 1^{+}} f (x)$
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Exercises 53
In the following exercises, use the following graph of the function $y = f (x)$ to find the values, if possible. Estimate when necessary.
$lim_{x \to 2} f (x)$
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Exercises 55
In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 0^{-}} f (x)$
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Exercises 57
In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 0} f (x)$
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Exercises 59
In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to - 2^{-}} f (x)$
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Exercises 61
In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to - 2} f (x)$
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Exercises 63
In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 2^{+}} f (x)$
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Exercises 65
In the following exercises, use the graph of the function $y = g (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 0^{-}} g (x)$
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Exercises 67
In the following exercises, use the graph of the function $y = g (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 0} g (x)$
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Exercises 69
In the following exercises, use the graph of the function $y = h (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 0^{+}} h (x)$
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Exercises 71
In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 0^{-}} f (x)$
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Exercises 73
In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 0} f (x)$
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Exercises 75
In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.
$lim_{x \to 2} f (x)$
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Exercises 77
In the following exercises, sketch the graph of a function with the given properties.
$A s x \to - \infty, f (x) \to 0, lim_{x \to - 1^{-}} f (x) = - \infty,$
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Exercises 79
In the following exercises, sketch the graph of a function with the given properties.
$A s x \to - \infty, f (x) \to 2, lim_{x \to - 2} f (x) = - \infty,$ $A s x \to \infty, f (x) \to 2, f (0) = 0$
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Exercises 81
In the following exercises, sketch the graph of a function with the given properties.
Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, x, is shown here. We are mainly interested in the location of the front of the shock, labeled $x_{SF}$ in the diagram.
Evaluate $lim_{x \to x_{S F}^{+}} ρ (x) .$
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Review Exercises 213
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.
$lim_{x \to 2} \frac{2 x ^{2} - 3 x - 2}{x - 2}$
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Review Exercises 215
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.
$lim_{x \to 3} \frac{x ^{3} - 2 x ^{2} - 1}{3 x - 2}$
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Review Exercises 217
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.
$lim_{x \to - 5} \frac{x ^{2} + 25}{x + 5}$
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Review Exercises 219
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.
$lim_{x \to 1} \frac{x ^{2} - 1}{x ^{3} - 1}$
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Review Exercises 221
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.
$lim_{x \to 4} \frac{4 - x}{x - 2}$
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Key terms · 4

infinite limit: A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a
intuitive definition of the limit: If all values of the function $f (x)$ approach the real number L as the values of $x (\ne a)$ approach a, $f (x)$ approaches L
one-sided limit: A one-sided limit of a function is a limit taken from either the left or the right
vertical asymptote: A function has a vertical asymptote at $x = a$ if the limit as x approaches a from the right or left is infinite

Learn 2.2 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 2.2 in the textbook · This section on openstax.org

The Limit of a Function: practice problems

31 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 31

For the following exercises, consider the function $f (x) = \frac{x ^{2} - 1}{∣ x - 1∣} .$

What do your results in the preceding exercise indicate about the two-sided limit $lim_{x \to 1} f (x)?$ Explain your response.

Exercises 33

For the following exercises, consider the function $f (x) = (1 + x)^{1 / x} .$

What does the table of values in the preceding exercise indicate about the function $f (x) = (1 + x)^{1 / x} ?$

Exercises 35

In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.

[T] $lim_{x \to 0} \frac{s i n 2 x}{x}; \pm 0.1, \pm 0.01, \pm 0.001, \pm .0001$

Exercises 37

In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.

Use the preceding two exercises to conjecture (guess) the value of the following limit: $lim_{x \to 0} \frac{s i n a x}{x}$ for a, a positive real value.

Exercises 39

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.

$lim_{x \to 1} (1 - 2 x)$

1 - 2 x

1 - 2 x

Exercises 41

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.

$lim_{z \to 0} \frac{z - 1}{z ^{2} ( z + 3 )}$

Exercises 43

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.

$lim_{x \to 2} \frac{1 - \frac{2}{x}}{x ^{2} - 4}$

Exercises 45

[T] In the following exercises, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?

$lim_{α \to 0^{+}} \frac{1}{α} cos (\frac{π}{α})$

Exercises 47

In the following exercises, consider the graph of the function $y = f (x)$ shown here. Which of the statements about $y = f (x)$ are true and which are false? Explain why a statement is false.

A graph of a piecewise function with three segments and a point. The first segment is a curve opening upward with vertex at (-8, -6). This vertex is an open circle, and there is a closed circle instead at (-8, -3). The segment ends at (-2,3), where there is a closed circle. The second segment stretches up asymptotically to infinity along x=-2, changes direction to increasing at about (0,1.25), increases until about (2.25, 3), and decreases until (6,2), where there is an open circle. The last segment starts at (6,5), increases slightly, and then decreases into quadrant four, crossing the x axis at (10,0). All of the changes in direction are smooth curves.

$lim_{x \to - 2^{+}} f (x) = 3$

Exercises 49

In the following exercises, consider the graph of the function $y = f (x)$ shown here. Which of the statements about $y = f (x)$ are true and which are false? Explain why a statement is false.

$lim_{x \to 6} f (x) = 5$

Exercises 51

In the following exercises, use the following graph of the function $y = f (x)$ to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first segment exists for x <=1, and the second segment exists for x > 1. The first segment is linear with a slope of 1 and goes through the origin. Its endpoint is a closed circle at (1,1). The second segment is also linear with a slope of -1. It begins with the open circle at (1,2).

$lim_{x \to 1^{+}} f (x)$

Exercises 53

In the following exercises, use the following graph of the function $y = f (x)$ to find the values, if possible. Estimate when necessary.

$lim_{x \to 2} f (x)$

Exercises 55

In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first is a linear function for x < 0. There is an open circle at (0,1), and its slope is -1. The second segment is the right half of a parabola opening upward. Its vertex is a closed circle at (0, -4), and it goes through the point (2,0).

$lim_{x \to 0^{-}} f (x)$

Exercises 57

In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.

$lim_{x \to 0} f (x)$

Exercises 59

In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with three segments, all linear. The first exists for x < -2, has a slope of 1, and ends at the open circle at (-2, 0). The second exists over the interval [-2, 2], has a slope of -1, goes through the origin, and has closed circles at its endpoints (-2, 2) and (2,-2). The third exists for x>2, has a slope of 1, and begins at the open circle (2,2).

$lim_{x \to - 2^{-}} f (x)$

Exercises 61

In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.

$lim_{x \to - 2} f (x)$

Exercises 63

In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.

$lim_{x \to 2^{+}} f (x)$

Exercises 65

In the following exercises, use the graph of the function $y = g (x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first exists for x>=0 and is the left half of an upward opening parabola with vertex at the closed circle (0,3). The second exists for x>0 and is the right half of a downward opening parabola with vertex at the open circle (0,0).

$lim_{x \to 0^{-}} g (x)$

Exercises 67

In the following exercises, use the graph of the function $y = g (x)$ shown here to find the values, if possible. Estimate when necessary.

$lim_{x \to 0} g (x)$

Exercises 69

In the following exercises, use the graph of the function $y = h (x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a function with two curves approaching 0 from quadrant 1 and quadrant 3. The curve in quadrant one appears to be the top half of a parabola opening to the right of the y axis along the x axis with vertex at the origin. The curve in quadrant three appears to be the left half of a parabola opening downward with vertex at the origin.

$lim_{x \to 0^{+}} h (x)$

Exercises 71

In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.

A graph with a curve and a point. The point is a closed circle at (0,-2). The curve is part of an upward opening parabola with vertex at (1,-1). It exists for x > 0, and there is a closed circle at the origin.

$lim_{x \to 0^{-}} f (x)$

Exercises 73

In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.

$lim_{x \to 0} f (x)$

Exercises 75

In the following exercises, use the graph of the function $y = f (x)$ shown here to find the values, if possible. Estimate when necessary.

$lim_{x \to 2} f (x)$

Exercises 77

In the following exercises, sketch the graph of a function with the given properties.

$A s x \to - \infty, f (x) \to 0, lim_{x \to - 1^{-}} f (x) = - \infty,$

Exercises 79

In the following exercises, sketch the graph of a function with the given properties.

$A s x \to - \infty, f (x) \to 2, lim_{x \to - 2} f (x) = - \infty,$ $A s x \to \infty, f (x) \to 2, f (0) = 0$

Exercises 81

In the following exercises, sketch the graph of a function with the given properties.

Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, x, is shown here. We are mainly interested in the location of the front of the shock, labeled $x_{SF}$ in the diagram.

A graph in quadrant one of the density of a shockwave with three labeled points: p1 and p2 on the y axis, with p1 > p2, and xsf on the x axis. It consists of y= p1 from 0 to xsf, x = xsf from y= p1 to y=p2, and y=p2 for values greater than or equal to xsf.

Evaluate $lim_{x \to x_{S F}^{+}} ρ (x) .$

Review Exercises 213

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$lim_{x \to 2} \frac{2 x ^{2} - 3 x - 2}{x - 2}$

Review Exercises 215

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$lim_{x \to 3} \frac{x ^{3} - 2 x ^{2} - 1}{3 x - 2}$

Review Exercises 217

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$lim_{x \to - 5} \frac{x ^{2} + 25}{x + 5}$

Review Exercises 219

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$lim_{x \to 1} \frac{x ^{2} - 1}{x ^{3} - 1}$

Review Exercises 221

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$lim_{x \to 4} \frac{4 - x}{x - 2}$

Key terms · 4

infinite limit: A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a
intuitive definition of the limit: If all values of the function $f (x)$ approach the real number L as the values of $x (\ne a)$ approach a, $f (x)$ approaches L
one-sided limit: A one-sided limit of a function is a limit taken from either the left or the right
vertical asymptote: A function has a vertical asymptote at $x = a$ if the limit as x approaches a from the right or left is infinite

Learn 2.2 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 2.2 in the textbook · This section on openstax.org

x	$\frac{s i n 2 x}{x}$		x	$\frac{s i n 2 x}{x}$
−0.1	a.		0.1	e.
−0.01	b.	0.01	f.
−0.001	c.	0.001	g.
−0.0001	d.	0.0001	h.

x	$\frac{1 - \frac{2}{x}}{x ^{2} - 4}$		x	$\frac{1 - \frac{2}{x}}{x ^{2} - 4}$
1.9	a.		2.1	e.
1.99	b.	2.01	f.
1.999	c.	2.001	g.
1.9999	d.	2.0001	h.

$α$	$\frac{1}{α} cos (\frac{π}{α})$
0.1	a.
0.01	b.
0.001	c.
0.0001	d.