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lim

Selecting Procedures for
Determining Limits

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

AP Calculus AB

What Is a Limit?

A limit describes the value a function \( f(x) \) approaches as \( x \) gets closer and closer to some number \( c \). We write this as:

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

Why Procedure Selection Matters

Not every limit can be solved the same way. Plugging in \( x = c \) sometimes gives \( \frac{0}{0} \), which is indeterminate: it tells you nothing about the actual limit.

✓ Determinate
Direct substitution yields a real number like \( \frac{7}{3} \). You are done.
★ Indeterminate
Substitution yields \( \frac{0}{0} \). More algebra is needed to resolve the limit.
✕ Non-existent
Substitution yields \( \frac{k}{0} \) with \( k \neq 0 \). The limit is \( \pm\infty \) or DNE.

The Limit Decision Flowchart

Try Direct Substitution What does f(c) give you? Limit = f(c) Real number Limit is ±∞ or DNE k/0, k ≠ 0 Indeterminate: 0/0 Choose an algebraic technique: Factor & Cancel Conjugate / Rationalize Trig Identities Common Denominator Then substitute again to find L

Procedure 1: Direct Substitution

Direct substitution is always your first step. If \( f(x) \) is a polynomial, rational function with no zero denominator, exponential, or basic trig function at \( x = c \), simply compute \( f(c) \).

Recognizing the Indeterminate Form

When direct substitution gives \( \frac{0}{0} \), the limit is indeterminate: it could be any real number, \( \pm\infty \), or DNE. You must transform the expression algebraically.

hole at x = 2 x x y

Procedure 2: Factor and Cancel

When you get \( \frac{0}{0} \), look for a common factor in numerator and denominator that causes both to be zero at \( x = c \). Factor it out, cancel, then substitute again.

1
Factor the numerator and denominator completely.
2
Cancel the common factor \( (x - c) \).
3
Substitute \( x = c \) into the simplified expression.

Procedure 3: Rationalize (Conjugate)

When \( \frac{0}{0} \) arises and the expression contains a square root, multiply numerator and denominator by the conjugate to eliminate the radical.

1
Identify the term with \( \sqrt{\phantom{x}} \). Form its conjugate.
2
Multiply top and bottom by the conjugate. Use \( (a-b)(a+b) = a^2 - b^2 \).
3
Simplify, cancel the common factor, then substitute.

Procedure 4: Special Trig Limits

Two fundamental trig limits are needed throughout calculus. Memorize them and manipulate expressions to match these forms.

Sine Limit
$$\lim_{x \to 0}\frac{\sin x}{x} = 1$$
Cosine Limit
$$\lim_{x \to 0}\frac{1 - \cos x}{x} = 0$$

Procedure 5: Combine Fractions

When a limit expression contains a complex fraction (fraction within a fraction), combine fractions into a single ratio, then simplify.

1
Combine the numerator fractions: \(\displaystyle \frac{1}{x+3}-\frac{1}{3}=\frac{3-(x+3)}{3(x+3)}=\frac{-x}{3(x+3)}\)
2
Divide by \( x \): \(\displaystyle \frac{-x}{3(x+3)}\cdot\frac{1}{x}=\frac{-1}{3(x+3)}\)
3
Substitute \( x = 0 \): \(\displaystyle \frac{-1}{3(3)}=\) \(-\frac{1}{9}\)

\(\frac{0}{0}\) vs. \(\frac{k}{0}\): A Critical Distinction

\(\frac{0}{0}\) Indeterminate \(\frac{k}{0}\), \(k \neq 0\)
Meaning Both numerator and denominator approach 0 Numerator approaches nonzero; denominator approaches 0
Limit? May exist: need algebraic work \(\pm\infty\) or DNE (vertical asymptote)
Next Step Factor, rationalize, trig identity, or combine fractions Analyze sign of numerator/denominator from left and right
Example \(\displaystyle\frac{x^2-1}{x-1}\) at \(x=1\) \(\displaystyle\frac{5}{x-2}\) at \(x=2\)

How to Select the Right Procedure

1
Always try direct substitution first. Plug \( x = c \) into \( f(x) \).
2
If you get a real number, that is the limit. Stop.
3
If you get \(\frac{0}{0}\), examine the expression's form: polynomial → factor; radical → conjugate; trig → special limit; nested fractions → combine.
4
Simplify and cancel the common factor \( (x - c) \).
5
Substitute again. If still indeterminate, repeat or try another technique.

Example 1: Factoring

1
Substitute \( x = -1 \): \(\displaystyle \frac{(-1)^2+3(-1)+2}{-1+1}=\frac{1-3+2}{0}=\frac{0}{0}\) → indeterminate
2
Factor: \(\displaystyle \frac{(x+1)(x+2)}{x+1}\)
3
Cancel \( (x+1) \) and substitute: \(\displaystyle \lim_{x \to -1}(x+2) = -1+2 =\) 1

Example 2: Conjugate

1
Substitute \( x = 9 \): \(\frac{\sqrt{9}-3}{9-9}=\frac{0}{0}\) → indeterminate
2
Multiply by conjugate: \(\displaystyle \frac{\sqrt{x}-3}{x-9}\cdot\frac{\sqrt{x}+3}{\sqrt{x}+3}=\frac{x-9}{(x-9)(\sqrt{x}+3)}\)
3
Cancel and substitute: \(\displaystyle \frac{1}{\sqrt{x}+3}\bigg|_{x=9}=\frac{1}{3+3}=\) \(\frac{1}{6}\)

Example 3: Trig Manipulation

1
Substitute \( x = 0 \): \(\frac{\sin 0}{\sin 0}=\frac{0}{0}\) → indeterminate
2
Rewrite to use \(\frac{\sin u}{u}\to 1\): \(\displaystyle\frac{\sin(5x)}{5x}\cdot\frac{2x}{\sin(2x)}\cdot\frac{5x}{2x}\)
3
Each ratio → 1: \(\displaystyle 1 \cdot 1 \cdot \frac{5}{2}=\) \(\frac{5}{2}\)

Common Mistakes to Avoid

✕ Skipping Substitution
Jumping to factoring before checking if direct substitution works wastes time and invites errors.
✕ Treating \(\frac{0}{0}\) as Zero
Writing \(\frac{0}{0}=0\) or \(\frac{0}{0}=1\) is wrong. It is indeterminate: the limit could be anything.
✕ Confusing \(\frac{0}{0}\) with \(\frac{k}{0}\)
\(\frac{k}{0}\) (with \(k\neq 0\)) means a vertical asymptote, not an algebraic simplification opportunity.

Summary: Procedure Selection Guide

Expression Clue Procedure Example Pattern
Polynomial or simple rational Direct substitution \(\frac{x^2+1}{x+5}\) at \(x=2\)
Factorable polynomial giving \(\frac{0}{0}\) Factor and cancel \(\frac{x^2-9}{x-3}\) at \(x=3\)
Square root in expression Conjugate / Rationalize \(\frac{\sqrt{x}-2}{x-4}\) at \(x=4\)
\(\sin\) or \(\cos\) with \(x \to 0\) Special trig limits \(\frac{\sin(3x)}{x}\) at \(x=0\)
Nested or compound fractions Combine fractions \(\frac{\frac{1}{x}-\frac{1}{a}}{x-a}\) at \(x=a\)

Question 1 of 5

What is \(\displaystyle\lim_{x \to 4}\frac{x^2 - 16}{x - 4}\)?

Question 2 of 5

Which procedure is most appropriate for evaluating \(\displaystyle\lim_{x \to 0}\frac{\sqrt{x+1}-1}{x}\)?

Question 3 of 5

Evaluate \(\displaystyle\lim_{x \to 0}\frac{\sin(7x)}{3x}\).

Question 4 of 5

If substituting \( x = c \) into \( f(x) \) yields \(\frac{5}{0}\), what can you conclude?

Question 5 of 5

Evaluate \(\displaystyle\lim_{x \to 3}\frac{\frac{1}{x}-\frac{1}{3}}{x - 3}\).