epsilon-delta definition of the limit

$\lim_{x \to a} f (x) = L$ if for every $\varepsilon > 0,$ there exists a $\delta > 0$ such that if $0 < | x - a | < \delta,$ then $| f (x) - L | < \varepsilon$

If a and b are any real numbers, then $| a + b | \le | a | + | b |$

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Calculus Volume 1 · 2.5 The Precise Definition of a Limit

The Precise Definition of a Limit: practice problems

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

Exercises 177
In the following exercises, write the appropriate $ε$ - $δ$ definition for each of the given statements.
$lim_{t \to b} g (t) = M$
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Exercises 179
In the following exercises, write the appropriate $ε$ - $δ$ definition for each of the given statements.
$lim_{x \to a} φ (x) = A$
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Exercises 181
The following graph of the function f satisfies $lim_{x \to 2} f (x) = 2.$ In the following exercises, determine a value of $δ > 0$ that satisfies each statement.
If $0 < ∣ x - 2∣ < δ,$ then
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Exercises 183
The following graph of the function f satisfies $lim_{x \to 3} f (x) = - 1.$ In the following exercises, determine a value of $δ > 0$ that satisfies each statement.
If $0 < ∣ x - 3∣ < δ,$ then
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Exercises 185
The following graph of the function f satisfies $lim_{x \to 3} f (x) = 2.$ In the following exercises, for each value of ε, find a value of $δ > 0$ such that the precise definition of limit holds true.
$ε = 3$
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Exercises 187
[T] In the following exercises, use a graphing calculator to find a number $δ$ such that the statements hold true.
$∣ x - 4 - 2∣ < 0.1, whenever ∣ x - 8∣ < δ$
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Exercises 189
In the following exercises, use the precise definition of limit to prove the given limits.
$lim_{x \to 3} \frac{x ^{2} - 9}{x - 3} = 6$
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Exercises 191
In the following exercises, use the precise definition of limit to prove the given limits.
$lim_{x \to 0} x^{4} = 0$
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Exercises 193
In the following exercises, use the precise definition of limit to prove the given one-sided limits.
$lim_{x \to 5^{-}} 5 - x = 0$
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Exercises 195
In the following exercises, use the precise definition of limit to prove the given one-sided limits.
$lim_{x \to 1^{-}} f (x) = 3, where f (x) = {5 x - 2, if x < 1 7 x - 1, if x \geq 1 .$
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Exercises 197
In the following exercises, use the precise definition of limit to prove the given infinite limits.
$lim_{x \to - 1} \frac{3}{( x + 1 ) ^{2}} = \infty$
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Exercises 199
In the following exercises, use the precise definition of limit to prove the given infinite limits.
An engineer is using a machine to cut a flat square of Aerogel of area 144 cm². If there is a maximum error tolerance in the area of 8 cm², how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to $δ,$ ε, a, and L?
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Exercises 201
In the following exercises, use the precise definition of limit to prove the given infinite limits.
Using precise definitions of limits, prove that $lim_{x \to 0} f (x)$ does not exist, given that $f (x)$ is the ceiling function. (Hint: Try any $δ < 1 .)$
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Exercises 203
In the following exercises, use the precise definition of limit to prove the given infinite limits.
Using precise definitions of limits, determine $lim_{x \to 0} f (x)$ for $f (x) = {x if x is rational 0 if x is irrational .$ (: Break into two cases, rational and irrational.)
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Exercises 205
For the following exercises, suppose that $lim_{x \to a} f (x) = L$ and $lim_{x \to a} g (x) = M$ both exist. Use the precise definition of limits to prove the following limit laws:
$lim_{x \to a} (f (x) + g (x)) = L + M$
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Exercises 207
For the following exercises, suppose that $lim_{x \to a} f (x) = L$ and $lim_{x \to a} g (x) = M$ both exist. Use the precise definition of limits to prove the following limit laws:
$lim_{x \to a} [f (x) g (x)] = L M .$ (:
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Review Exercises 229
In the following exercises, use the precise definition of limit to prove the limit.
$lim_{x \to 0} x^{3} = 0$
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Key terms · 2

epsilon-delta definition of the limit: $\lim_{x \to a} f (x) = L$ if for every $\varepsilon > 0,$ there exists a $\delta > 0$ such that if $0 < | x - a | < \delta,$ then $| f (x) - L | < \varepsilon$
triangle inequality: If a and b are any real numbers, then $| a + b | \le | a | + | b |$

Learn 2.5 on the whiteboard

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

Read 2.5 in the textbook · This section on openstax.org

The Precise Definition of a Limit: practice problems

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

Exercises 177

In the following exercises, write the appropriate $ε$ - $δ$ definition for each of the given statements.

$lim_{t \to b} g (t) = M$

Exercises 179

In the following exercises, write the appropriate $ε$ - $δ$ definition for each of the given statements.

$lim_{x \to a} φ (x) = A$

Exercises 181

The following graph of the function f satisfies $lim_{x \to 2} f (x) = 2.$ In the following exercises, determine a value of $δ > 0$ that satisfies each statement.

A function drawn in quadrant one for x > 0. It is an increasing concave up function, with points approximately (0,0), (1, .5), (2,2), and (3,4).

If $0 < ∣ x - 2∣ < δ,$ then

Exercises 183

The following graph of the function f satisfies $lim_{x \to 3} f (x) = - 1.$ In the following exercises, determine a value of $δ > 0$ that satisfies each statement.

A graph of a decreasing linear function, with points (0,2), (1,1), (2,0), (3,-1), (4,-2), and so on for x >= 0.

If $0 < ∣ x - 3∣ < δ,$ then

Exercises 185

The following graph of the function f satisfies $lim_{x \to 3} f (x) = 2.$ In the following exercises, for each value of ε, find a value of $δ > 0$ such that the precise definition of limit holds true.

A graph of an increasing linear function intersecting the x axis at about (2.25, 0) and going through the points (3,2) and, approximately, (1,-5) and (4,5).

$ε = 3$

Exercises 187

[T] In the following exercises, use a graphing calculator to find a number $δ$ such that the statements hold true.

$∣ x - 4 - 2∣ < 0.1, whenever ∣ x - 8∣ < δ$

Exercises 189

In the following exercises, use the precise definition of limit to prove the given limits.

$lim_{x \to 3} \frac{x ^{2} - 9}{x - 3} = 6$

Exercises 191

In the following exercises, use the precise definition of limit to prove the given limits.

$lim_{x \to 0} x^{4} = 0$

Exercises 193

In the following exercises, use the precise definition of limit to prove the given one-sided limits.

$lim_{x \to 5^{-}} 5 - x = 0$

Exercises 195

In the following exercises, use the precise definition of limit to prove the given one-sided limits.

$lim_{x \to 1^{-}} f (x) = 3, where f (x) = {5 x - 2, if x < 1 7 x - 1, if x \geq 1 .$

Exercises 197

In the following exercises, use the precise definition of limit to prove the given infinite limits.

$lim_{x \to - 1} \frac{3}{( x + 1 ) ^{2}} = \infty$

Exercises 199

In the following exercises, use the precise definition of limit to prove the given infinite limits.

An engineer is using a machine to cut a flat square of Aerogel of area 144 cm². If there is a maximum error tolerance in the area of 8 cm², how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to $δ,$ ε, a, and L?

Exercises 201

In the following exercises, use the precise definition of limit to prove the given infinite limits.

Using precise definitions of limits, prove that $lim_{x \to 0} f (x)$ does not exist, given that $f (x)$ is the ceiling function. (Hint: Try any $δ < 1 .)$

Exercises 203

In the following exercises, use the precise definition of limit to prove the given infinite limits.

Using precise definitions of limits, determine $lim_{x \to 0} f (x)$ for $f (x) = {x if x is rational 0 if x is irrational .$ (: Break into two cases, rational and irrational.)

Exercises 205

For the following exercises, suppose that $lim_{x \to a} f (x) = L$ and $lim_{x \to a} g (x) = M$ both exist. Use the precise definition of limits to prove the following limit laws:

$lim_{x \to a} (f (x) + g (x)) = L + M$

Exercises 207

For the following exercises, suppose that $lim_{x \to a} f (x) = L$ and $lim_{x \to a} g (x) = M$ both exist. Use the precise definition of limits to prove the following limit laws:

$lim_{x \to a} [f (x) g (x)] = L M .$ (:

Review Exercises 229

In the following exercises, use the precise definition of limit to prove the limit.

$lim_{x \to 0} x^{3} = 0$

Key terms · 2

epsilon-delta definition of the limit: $\lim_{x \to a} f (x) = L$ if for every $\varepsilon > 0,$ there exists a $\delta > 0$ such that if $0 < | x - a | < \delta,$ then $| f (x) - L | < \varepsilon$
triangle inequality: If a and b are any real numbers, then $| a + b | \le | a | + | b |$

Learn 2.5 on the whiteboard

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

Read 2.5 in the textbook · This section on openstax.org