Calculus Practice

Limits Practice

Practice evaluating limits using direct substitution, factoring, rationalizing, one-sided limits, infinite limits, and continuity.

Start Practicing Review Limits

Try each problem first, then click Show solution to check your reasoning and see the complete step-by-step solution.

Keep These Ideas Nearby

Limits Essentials

Limit Meaning

\[ \lim_{x\to a}f(x)=L \]

\(f(x)\) approaches \(L\) as \(x\) approaches \(a\).

One-Sided Limits

\[ \lim_{x\to a^-}f(x), \qquad \lim_{x\to a^+}f(x) \]

Approach from the left and from the right.

Continuity

\[ \lim_{x\to a}f(x)=f(a) \]

Required when \(f\) is continuous at \(x=a\).

Download Practice PDF Review Limits

Level 1

Understanding Limits

Start with notation and the difference between function values and limits.

Problem 1

Read Limit Notation

What does \[ \lim_{x\to3}f(x)=7 \] mean?

\(f(3)=7\)
\(f(x)\) approaches \(7\) as \(x\) approaches \(3\)
\(x\) approaches \(7\)

Step-by-Step Solution

A limit tells us the value that the function approaches as the input approaches a particular number.

\[ \lim_{x\to3}f(x)=7 \]

This does not necessarily mean that \(f(3)=7\).

Answer: \(f(x)\) approaches \(7\) as \(x\) approaches \(3\).

Problem 2

Function Value vs Limit

Suppose \[ f(2)=1 \] but \[ \lim_{x\to2}f(x)=5. \] Which statement is correct?

The limit must equal \(1\)
The function approaches \(5\), even though \(f(2)=1\)
The limit does not exist

Step-by-Step Solution

The value \(f(2)\) tells us what happens exactly at \(x=2\).

The limit describes what happens near \(x=2\).

\[ f(2)=1 \] \[ \lim_{x\to2}f(x)=5 \]
Answer: The function approaches \(5\), even though \(f(2)=1\).

Problem 3

Estimate from a Table

Use the table to estimate \[ \lim_{x\to2}f(x). \]

\(x\) \(f(x)\)
1.9 3.9
1.99 3.99
2.01 4.01
2.1 4.1

Step-by-Step Solution

As \(x\) approaches \(2\) from both sides, the values of \(f(x)\) approach \(4\).

\[ 3.9,\; 3.99,\; 4.01,\; 4.1 \] all cluster near \(4\).
Answer: \[ \boxed{4} \]

Level 2

Direct Substitution

Evaluate limits where substitution works immediately, and recognize when an indeterminate form appears.

Problem 4

Direct Substitution

Evaluate: \[ \lim_{x\to3} \left(x^2+2x-1\right) \]

Step-by-Step Solution

Substitute \(x=3\).

\[ 3^2+2(3)-1 \] \[ =9+6-1 \] \[ =14 \]
Answer: \[ \boxed{14} \]

Problem 5

Rational Function

Evaluate: \[ \lim_{x\to2} \frac{x+4}{x+1} \]

Step-by-Step Solution

Direct substitution gives a nonzero denominator, so substitution works.

\[ \frac{2+4}{2+1} = \frac63 = 2 \]
Answer: \[ \boxed{2} \]

Problem 6

Recognize \(0/0\)

What happens when direct substitution is used on \[ \lim_{x\to4} \frac{x^2-16}{x-4}? \]

\[ 4 \]
\[ \frac00 \]
\[ \infty \]

Step-by-Step Solution

Substitute \(x=4\).

\[ \frac{4^2-16}{4-4} = \frac{16-16}{0} = \frac00 \]

This is an indeterminate form, so more algebra is required.

Answer: \[ \boxed{\frac00} \]

Level 3

Algebraic Limits

Use factoring and conjugates to remove indeterminate forms.

Problem 7

Factoring

Evaluate: \[ \lim_{x\to3} \frac{x^2-9}{x-3} \]

Step-by-Step Solution

Direct substitution gives \(0/0\), so factor the numerator.

\[ x^2-9 = (x-3)(x+3) \] \[ \frac{(x-3)(x+3)}{x-3} = x+3 \]

Now substitute \(x=3\).

\[ 3+3=6 \]
Answer: \[ \boxed{6} \]

Problem 8

Factor a Quadratic

Evaluate: \[ \lim_{x\to2} \frac{x^2-x-2}{x-2} \]

Step-by-Step Solution

Factor the numerator.

\[ x^2-x-2 = (x-2)(x+1) \] \[ \frac{(x-2)(x+1)}{x-2} = x+1 \]

Substitute \(x=2\).

\[ 2+1=3 \]
Answer: \[ \boxed{3} \]

Problem 9

Rationalize

Evaluate: \[ \lim_{x\to4} \frac{\sqrt{x}-2}{x-4} \]

Step-by-Step Solution

Direct substitution gives \(0/0\), so multiply by the conjugate.

\[ \frac{\sqrt{x}-2}{x-4} \cdot \frac{\sqrt{x}+2}{\sqrt{x}+2} \] \[ = \frac{x-4} {(x-4)(\sqrt{x}+2)} \] \[ = \frac{1}{\sqrt{x}+2} \]

Now substitute \(x=4\).

\[ \frac{1}{2+2} = \frac14 \]
Answer: \[ \boxed{\frac14} \]

Level 4

One-Sided and Infinite Limits

Compare behavior from each side and identify when a two-sided limit does not exist.

Problem 10

One-Sided Limits

Suppose \[ \lim_{x\to2^-}f(x)=4 \] and \[ \lim_{x\to2^+}f(x)=4. \] Find \[ \lim_{x\to2}f(x). \]

Step-by-Step Solution

A two-sided limit exists when the left-hand and right-hand limits agree.

\[ \lim_{x\to2^-}f(x) = \lim_{x\to2^+}f(x) = 4 \]
Answer: \[ \boxed{4} \]

Problem 11

DNE

Suppose \[ \lim_{x\to5^-}f(x)=2 \] and \[ \lim_{x\to5^+}f(x)=8. \] Find \[ \lim_{x\to5}f(x). \]

Step-by-Step Solution

The left-hand and right-hand limits do not agree.

\[ 2\neq8 \]

Therefore the two-sided limit does not exist.

Answer: \[ \boxed{\text{DNE}} \]

Problem 12

Infinite Limit

Evaluate: \[ \lim_{x\to0^+}\frac1x \]

Step-by-Step Solution

Approaching zero from the right means using small positive numbers.

\[ \frac{1}{0.1}=10 \] \[ \frac{1}{0.01}=100 \] \[ \frac{1}{0.001}=1000 \]

The values increase without bound.

Answer: \[ \boxed{\infty} \]

Problem 13

Vertical Asymptote

For \[ f(x)=\frac{1}{x-3}, \] what vertical line is a vertical asymptote?

Step-by-Step Solution

A vertical asymptote often occurs where the denominator becomes zero and the function becomes unbounded.

\[ x-3=0 \] \[ x=3 \]
Answer: \[ \boxed{x=3} \]

Level 5

Mixed Limits and Continuity

Choose the appropriate strategy and connect limits to continuity.

Problem 14

Choose a Strategy

Evaluate: \[ \lim_{x\to5} \frac{x^2-25}{x-5} \]

Step-by-Step Solution

Direct substitution gives \(0/0\), so factor.

\[ x^2-25 = (x-5)(x+5) \] \[ \frac{(x-5)(x+5)}{x-5} = x+5 \] \[ 5+5=10 \]
Answer: \[ \boxed{10} \]

Problem 15

Continuity Test

Suppose \[ f(4)=7 \] and \[ \lim_{x\to4}f(x)=7. \] Is \(f\) continuous at \(x=4\)?

Step-by-Step Solution

Continuity at \(x=4\) requires:

  1. \(f(4)\) exists.
  2. \(\lim_{x\to4}f(x)\) exists.
  3. They are equal.
\[ \lim_{x\to4}f(x)=f(4)=7 \]
Answer: Yes, \(f\) is continuous at \(x=4\).

Problem 16

Discontinuity

Suppose \[ \lim_{x\to2}f(x)=6 \] but \[ f(2)=1. \] Is \(f\) continuous at \(x=2\)?

Step-by-Step Solution

Continuity requires the function value to equal the limit.

\[ \lim_{x\to2}f(x)=6 \] but \[ f(2)=1. \]

Since \(6\neq1\), the continuity condition fails.

Answer: No. The function is not continuous at \(x=2\).

Before You Finish

Limits Checklist

  1. Know that a limit describes what a function approaches.
  2. Keep \(f(a)\) separate from \(\lim\limits_{x\to a}f(x)\).
  3. Try direct substitution first.
  4. If substitution gives \(0/0\), simplify before evaluating.
  5. Factor polynomial expressions when a common factor can cancel.
  6. Use a conjugate when radicals cause an indeterminate form.
  7. Compare left-hand and right-hand limits for two-sided limits.
  8. Recognize infinite behavior near vertical asymptotes.
  9. For continuity, verify that the function value exists, the limit exists, and the two are equal.

Continue Learning

Related Calculus Resources

Review the full limits lesson or continue into derivatives, where limits become one of the central tools of calculus.

Limits Guide

Review limit notation, substitution, factoring, one-sided limits, and continuity.

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