Infinite Limits and Vertical Asymptotes

When functions shoot off to infinity at a specific point

Infinite Limits

Sometimes a function doesn't approach a finite value - it shoots off to infinity!

The Notation

$\lim_{x \to a} f(x) = \infty$

This means: as x approaches a, f(x) grows without bound (gets arbitrarily large).

$\lim_{x \to a} f(x) = -\infty$

This means: as x approaches a, f(x) becomes arbitrarily negative.

Important: When we write $= \infty$ , the limit does not exist (DNE) in the traditional sense. We're just being specific about how it doesn't exist.

Vertical Asymptotes

If $\lim_{x \to a} f(x) = \pm\infty$ , then x = a is a vertical asymptote.

The graph shoots up or down near this vertical line.

Common Cause: Division by Zero

The most common way to get infinite limits: denominator approaches 0 while numerator doesn't.

Example: $f(x) = \frac{1}{x - 2}$

As $x \to 2$ :

Numerator: 1 (constant)
Denominator: $x - 2 \to 0$

Result: $\frac{1}{\text{tiny number}} = \text{huge number}$

One-Sided Infinite Limits

We often need to check both sides because they might go different directions!

Example: $f(x) = \frac{1}{x - 2}$

From the left ( $x \to 2^-$ ):

When x = 1.9: $\frac{1}{1.9 - 2} = \frac{1}{-0.1} = -10$
When x = 1.99: $\frac{1}{1.99 - 2} = \frac{1}{-0.01} = -100$
When x = 1.999: $\frac{1}{-0.001} = -1000$

$\lim_{x \to 2^-} \frac{1}{x - 2} = -\infty$

From the right ( $x \to 2^+$ ):

When x = 2.1: $\frac{1}{2.1 - 2} = \frac{1}{0.1} = 10$
When x = 2.01: $\frac{1}{0.01} = 100$
When x = 2.001: $\frac{1}{0.001} = 1000$

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

The Sign Test

To determine if the limit is $+\infty$ or $-\infty$ :

Find where denominator = 0
Check a test point just to the left
Check a test point just to the right
Determine the sign of the function

Visual Interpretation

On a graph:

Vertical asymptote: Dashed vertical line at x = a
Left side: Graph shoots up ( $+\infty$ ) or down ( $-\infty$ )
Right side: Graph shoots up or down
The graph never touches the vertical asymptote

Example with Factoring

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

As $x \to 3$ :

Numerator: $3 + 1 = 4$ (positive)
Denominator: $(3 - 3)^2 = 0$

But $(x - 3)^2$ is always positive (it's squared!)

So from both sides: $\frac{\text{positive}}{\text{positive tiny}} = +\infty$

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

Both sides go to $+\infty$ !

📚 Practice Problems

1Problem 1medium

❓ Question:

Find $\lim_{x \to 1^-} \frac{2}{x - 1}$ and $\lim_{x \to 1^+} \frac{2}{x - 1}$

💡 Show Solution

Approaching from the left ( $x \to 1^-$ ):

Test with x = 0.9: $\frac{2}{0.9 - 1} = \frac{2}{-0.1} = -20$

The numerator is positive (2), the denominator is negative (approaching 0 from below).

$\lim_{x \to 1^-} \frac{2}{x - 1} = -\infty$

Approaching from the right ( $x \to 1^+$ ):

Test with x = 1.1: $\frac{2}{1.1 - 1} = \frac{2}{0.1} = 20$

The numerator is positive (2), the denominator is positive (approaching 0 from above).

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

Conclusion:

Left-hand limit: $-\infty$
Right-hand limit: $+\infty$
There is a vertical asymptote at x = 1

2Problem 2hard

❓ Question:

Find $\lim_{x \to -2} \frac{x}{(x + 2)^2}$ and describe the behavior

💡 Show Solution

Step 1: Check what happens at x = -2

Numerator: $x \to -2$ (negative) Denominator: $(x + 2)^2 \to 0$ (positive, because it's squared)

Step 2: Determine the sign

$\frac{\text{negative}}{\text{positive tiny}} = \text{large negative}$

Step 3: Check both sides

Since $(x + 2)^2$ is always positive (squared term), and the numerator x is negative near -2, the function will be negative on both sides.

From the left (x = -2.1): $\frac{-2.1}{(-2.1 + 2)^2} = \frac{-2.1}{0.01} = -210$

From the right (x = -1.9): $\frac{-1.9}{(-1.9 + 2)^2} = \frac{-1.9}{0.01} = -190$

Both sides go to $-\infty$ !

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

Behavior: There is a vertical asymptote at x = -2, and the graph approaches $-\infty$ from both sides.

3Problem 3easy

❓ Question:

Find lim(x→3⁺) 1/(x - 3)²

💡 Show Solution

Step 1: Analyze what happens as x→3⁺: x approaches 3 from the right (x > 3)

Step 2: Examine denominator: x - 3 approaches 0 from the positive side (x - 3)² is always positive and approaches 0

Step 3: Analyze the fraction: 1/(small positive number) = large positive number As denominator→0⁺, fraction→+∞

Step 4: Test values: x = 3.1: 1/(0.1)² = 1/0.01 = 100 x = 3.01: 1/(0.01)² = 10,000 Pattern: approaching +∞

Step 5: Conclusion: lim(x→3⁺) 1/(x - 3)² = +∞

Answer: +∞

4Problem 4medium

❓ Question:

Determine lim(x→-2⁻) (x + 1)/(x + 2)

💡 Show Solution

Step 1: Analyze as x→-2⁻: Approaching -2 from the left (x < -2)

Step 2: Evaluate numerator: x + 1 → -2 + 1 = -1 (negative)

Step 3: Evaluate denominator: x + 2 approaches 0 from the left (negative side)

Step 4: Determine sign: (negative)/(small negative) = large positive lim(x→-2⁻) (x + 1)/(x + 2) = +∞

Step 5: Verify with values: x = -2.1: (-1.1)/(-0.1) = 11 (positive) x = -2.01: (-1.01)/(-0.01) = 101 Approaching +∞ ✓

Answer: +∞

5Problem 5hard

❓ Question:

Find the vertical asymptote(s) of f(x) = (x - 1)/(x² - 4) and determine the behavior near each.

💡 Show Solution

Step 1: Find vertical asymptotes: Set denominator = 0 x² - 4 = 0 x = ±2

Step 2: Check if numerator ≠ 0 at these points: At x = 2: numerator = 2 - 1 = 1 ≠ 0 ✓ At x = -2: numerator = -2 - 1 = -3 ≠ 0 ✓ Both are vertical asymptotes

Step 3: Analyze behavior near x = 2: lim(x→2⁻): (positive)/(small negative) = -∞ lim(x→2⁺): (positive)/(small positive) = +∞

Step 4: Analyze behavior near x = -2: lim(x→-2⁻): (negative)/(small positive) = -∞ lim(x→-2⁺): (negative)/(small negative) = +∞

Answer: Vertical asymptotes at x = ±2 x = 2: -∞ from left, +∞ from right x = -2: -∞ from left, +∞ from right

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