Calculus I · Limits and Continuity · lesson

Vertical Asymptotes and One-Sided Sign Analysis

Concept

Learning objectives

Find vertical asymptote candidates, cancel removable factors before classifying behavior, and use a sign chart to determine each one-sided infinite limit.

Vertical Asymptotes and Sign Analysis

The line $x=a$ is a vertical asymptote if the function becomes unbounded on at least one side of $a$ .

For a rational function

f(x)=\frac{p(x)}{q(x)},

zeros of $q(x)$ are candidates. But first simplify common factors.

Common mistake

A denominator zero does not automatically create a vertical asymptote. If the zero factor cancels with the numerator, the graph may have a removable hole instead.

Guided walkthrough

Identify the sign before saying infinity

Evaluate

\lim_{x\to2^-}\frac1{x-2}.

Show worked solution

As $x\to2^-$ , $x-2$ is a tiny negative number. Dividing $1$ by a tiny negative number gives a very large negative number. Therefore,

\boxed{\lim_{x\to2^-}\frac1{x-2}=-\infty}.

Interactive checkinfinite-sign-01

Evaluate $\lim_{x\to1^+}\frac1{x-1}$ .

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Show hint

From the right, the denominator is positive and tiny.

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Method

One-Sided Sign Chart

To analyze $p(x)/q(x)$ near $x=a$ :

• Factor numerator and denominator. • Cancel any common factor, while remembering the corresponding hole. • Determine the sign of every remaining factor just left of $a$ . • Determine the sign just right of $a$ . • Combine signs and decide whether the magnitude grows without bound.

Worked example

A sign chart with several factors

Find both one-sided limits of

f(x)=\frac{x+2}{(x-3)(x+1)}

at $x=3$ .

Show worked solution

Near $x=3$ , the factors $x+2$ and $x+1$ are positive. Only $x-3$ changes sign.

From the left of $3$ :

x+2>0,\qquad x-3<0,\qquad x+1>0.

The quotient is negative. Its denominator approaches zero in magnitude, so

\boxed{\lim_{x\to3^-}f(x)=-\infty}.

From the right of $3$ :

x+2>0,\qquad x-3>0,\qquad x+1>0.

The quotient is positive, so

\boxed{\lim_{x\to3^+}f(x)=+\infty}.

Thus $x=3$ is a vertical asymptote.

Worked example

Even multiplicity

Analyze

g(x)=\frac{x-4}{(x+2)^2}

near $x=-2$ .

Show worked solution

Near $-2$ , the numerator $x-4$ is negative. The denominator $(x+2)^2$ is positive on both sides and approaches zero.

Therefore, on both sides the quotient is negative with unbounded magnitude:

\boxed{\lim_{x\to-2^-}g(x)=-\infty},

\boxed{\lim_{x\to-2^+}g(x)=-\infty}.

The two one-sided infinite behaviors agree, so it is also common to write

\lim_{x\to-2}g(x)=-\infty.

This does not mean the limit is a real number; it means both sides decrease without bound.

Worked example

Exam-level: hole versus asymptote

Find and classify every discontinuity of

h(x)=\frac{(x-1)(x+4)}{(x-1)(x-3)}.

Show worked solution

The original denominator is zero at $x=1$ and $x=3$ . Cancel the common factor for $x\ne1$ :

h(x)=\frac{x+4}{x-3}.

At $x=1$ , the simplified formula is finite:

\frac{1+4}{1-3}=-\frac52.

Therefore $x=1$ is a removable hole, not a vertical asymptote.

At $x=3$ , the remaining denominator is zero while the numerator is $7\ne0$ . Thus $x=3$ is a vertical asymptote.

For signs near $3$ , the numerator is positive. The denominator $x-3$ is negative from the left and positive from the right:

\lim_{x\to3^-}h(x)=-\infty, \qquad \lim_{x\to3^+}h(x)=+\infty.

After the explanation

Use the section idea

Reading lens

Is the function growing without bound near a finite input, or settling into end behavior as the input grows?

Mental model

Vertical asymptotes describe local blow-up near an excluded finite input; end-behavior asymptotes describe the long-run trend as inputs grow in magnitude.

Decision

Near a denominator zero, build a sign chart for each side; at infinity, compare dominant powers or divide to expose the lasting term.

Common trap

Do not merge positive and negative infinity, and remember that square roots produce absolute values when factoring a large squared input.

Check yourself

You understand the section when you can predict signs and asymptotes before doing detailed algebra, then verify them with the expression.

Source & rights

Original instruction with traceable references.

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