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\(\lim\)
\(x \to a\)

Selecting Procedures for
Determining Limits

Master the decision-making process for evaluating limits: direct substitution, factoring, rationalization, and more.

AP Calculus BC

What Is a Limit?

A limit describes the value a function \( f(x) \) approaches as \( x \) gets arbitrarily close to some value \( a \). We write:

The limit may exist even if \( f(a) \) is undefined. The key question is: what procedure should you use to find \( L \)?

Why Procedure Selection Matters

Not every limit can be solved the same way. Direct substitution sometimes gives \( \frac{0}{0} \), which is an indeterminate form: not a final answer, but a signal to try another technique.

✓ Determinate
Substitution gives a real number like \( \frac{7}{3} \). You are done.
★ Indeterminate
\( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Apply algebraic or analytic techniques.
✕ Undefined
\( \frac{k}{0} \) with \( k \neq 0 \). The limit is \( \pm\infty \) or DNE.

The Limit Landscape

\(x\)
\(y\)
 a = 2 L = 4 from left from right Limit exists at the hole

The Decision Flowchart

Try Direct Substitution Result?
Done! \(L\)
 real number
\(\pm\infty\) or
 DNE
\(\frac{k}{0}, k \neq 0\)
\(\frac{0}{0}\)
 Apply Algebraic Technique Factor Rationalize Trig Identity L'Hopital's

Direct Substitution Property

Always try direct substitution first. It is the fastest technique and works more often than students expect. Only move to other methods when substitution yields an indeterminate form.

Example
\( \lim_{x \to 3} (x^2 + 1) = 3^2 + 1 = 10 \)
Example
\( \lim_{x \to 0} \cos x = \cos 0 = 1 \)

Indeterminate Forms

When substitution yields one of these forms, the limit might still exist: the form just tells you more work is needed.

\( \frac{0}{0} \)
Most common. Factor, rationalize, or use L'Hopital's Rule.
\( \frac{\infty}{\infty} \)
Common at \( x \to \infty \). Divide by highest power or use L'Hopital's.
\( 0 \cdot \infty \)
Rewrite as a fraction to get \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
\( \infty - \infty \)
Combine fractions or factor to simplify first.

Indeterminate vs Undefined

Indeterminate Undefined (non-zero/zero)
Form \( \frac{0}{0},\; \frac{\infty}{\infty},\; 0 \cdot \infty \) \( \frac{k}{0} \) where \( k \neq 0 \)
Meaning Limit may exist: need more work Limit is \( \pm\infty \) or DNE
Next Step Factor, rationalize, L'Hopital, etc. Check left/right behavior for \( \pm\infty \)
Example \( \lim_{x\to 1}\frac{x^2-1}{x-1} = 2 \) \( \lim_{x\to 0^+}\frac{1}{x} = +\infty \)

How to Select Your Procedure

1
Substitute \( x = a \) directly into \( f(x) \). If you get a real number, that is the limit.
2
Identify the form. Is the result \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( \frac{k}{0} \), or another indeterminate form?
3
For \( \frac{0}{0} \): try factoring and canceling, rationalizing (conjugate), trig identities, or L'Hopital's Rule.
4
For \( \frac{\infty}{\infty} \): divide numerator and denominator by the highest power of \( x \), or apply L'Hopital's Rule.
5
Substitute again after simplifying. Repeat if a new indeterminate form appears.

Example 1: Factoring

1
Substitute: \( \frac{9-9}{3-3} = \frac{0}{0} \). Indeterminate.
2
Factor: \( \frac{(x-3)(x+3)}{x-3} = x+3 \) for \( x \neq 3 \).
3
Substitute again: \( 3 + 3 = \) 6. The limit is 6.

Example 2: Rationalization

1
Substitute: \( \frac{\sqrt{4}-2}{0} = \frac{0}{0} \). Indeterminate.
2
Multiply by conjugate: \( \frac{\sqrt{x+4}-2}{x} \cdot \frac{\sqrt{x+4}+2}{\sqrt{x+4}+2} = \frac{x+4-4}{x(\sqrt{x+4}+2)} = \frac{1}{\sqrt{x+4}+2} \)
3
Substitute: \( \frac{1}{\sqrt{4}+2} = \frac{1}{4} \). The limit is \( \frac{1}{4} \).

Example 3: Trig Identity

1
Substitute: \( \frac{\sin 0}{0} = \frac{0}{0} \). Indeterminate.
2
Use the standard result: \( \lim_{u\to 0}\frac{\sin u}{u} = 1 \). Rewrite: \( \frac{\sin(3x)}{x} = 3 \cdot \frac{\sin(3x)}{3x} \).
3
Evaluate: As \( x \to 0 \), \( 3x \to 0 \), so the expression becomes \( 3 \cdot 1 = \) 3.

Example 4: L'Hopital's Rule

1
Substitute: \( \frac{e^0 - 1}{0} = \frac{0}{0} \). Indeterminate: L'Hopital applies.
2
Differentiate top and bottom: \( \frac{d}{dx}[e^x - 1] = e^x \), \( \frac{d}{dx}[x] = 1 \).
3
New limit: \( \lim_{x \to 0} \frac{e^x}{1} = e^0 = \) 1.

Limits at Infinity

For rational functions as \( x \to \infty \), compare the degrees of numerator and denominator:

\(\deg N < \deg D\)
Limit is \( 0 \). The denominator grows faster.
\(\deg N = \deg D\)
Limit is the ratio of leading coefficients.
\(\deg N > \deg D\)
Limit is \( \pm\infty \). The numerator dominates.

Common Mistakes

✕ Stopping at \( \frac{0}{0} \)
\( \frac{0}{0} \) is not a final answer. It signals that you need to simplify and try again.
✕ Misusing L'Hopital's
L'Hopital's Rule only applies to \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Using it on \( \frac{0}{5} \) gives wrong answers.
✕ Canceling terms
You can only cancel common factors, not terms. For example, in \( \frac{x^2+1}{x+1} \), you cannot cancel the \(x\) terms or the \(1\)s.

Procedure Summary

Situation Technique When to Use
Real number result Direct substitution Always try first
\( \frac{0}{0} \) with polynomial Factor and cancel Common factor visible
\( \frac{0}{0} \) with radicals Conjugate multiply \( \sqrt{\phantom{x}} \) in expression
\( \frac{0}{0} \) with trig Trig identities \( \sin, \cos, \tan \) present
\( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) L'Hopital's Rule Other techniques fail
\( x \to \pm\infty \) rational Degree comparison Polynomial / polynomial

Question 1 of 5

What is the first step you should take when evaluating any limit?

Question 2 of 5

Evaluate: \( \displaystyle\lim_{x \to 5} \frac{x^2 - 25}{x - 5} \)

Question 3 of 5

Which technique is most appropriate for evaluating \( \displaystyle\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} \)?

Question 4 of 5

Evaluate: \( \displaystyle\lim_{x \to \infty} \frac{4x^3 + x}{2x^3 - 7} \)

Question 5 of 5

When can you apply L'Hopital's Rule?