Infinite Limits and Vertical Asymptotes 2.4

2.4: Infinite Limits

Introduction to Infinite Limits

Infinite limits describe the behavior of a function as the input approaches a certain value and the output increases or decreases without bound. These limits are closely related to the concept of vertical asymptotes, which are lines where the function grows arbitrarily large in magnitude.

• Infinite Limit: If $f(x)$ increases or decreases without bound as $x$ approaches $a$, we write $\lim_{x \to a} f(x) = \infty$ or $-\infty$.

• Vertical Asymptote: The line $x = a$ is a vertical asymptote of $f(x)$ if $f(x)$ approaches $\infty$ or $-\infty$ as $x$ approaches $a$ from either side.

• Does Not Exist (DNE): If the left and right limits are not equal or do not both approach infinity, the limit does not exist.

Graphical Interpretation

Vertical asymptotes are visible on the graph of a function as lines where the curve shoots up or down without bound. For example, the function $f(x)$ may have vertical asymptotes at $x = 1$ and $x = 3$ if:

• $\lim_{x \to 1^-} f(x) = \infty$ and $\lim_{x \to 1^+} f(x) = \infty$

• $\lim_{x \to 3^-} f(x) = -\infty$ and $\lim_{x \to 3^+} f(x) = \infty$

• If the left and right limits at $x = 3$ are not equal, $\lim_{x \to 3} f(x)$ does not exist (DNE).

Example: The vertical lines $x = 1$ and $x = 3$ are vertical asymptotes for the function $f(x)$ shown in the graph.

Infinite Limits for Rational Functions

Limits Involving Division by Zero

When evaluating limits of rational functions as $x$ approaches a value that makes the denominator zero, the function may approach infinity or negative infinity, depending on the sign of the numerator and denominator.

• Example: $f(x) = \frac{1}{x}$

• $\lim_{x \to 0^-} \frac{1}{x} = -\infty$

• $\lim_{x \to 0^+} \frac{1}{x} = \infty$

• $\lim_{x \to 0} \frac{1}{x}$ does not exist (DNE) because the left and right limits are not equal.

Evaluating Infinite Limits: Examples

To evaluate infinite limits, substitute values approaching the point of interest and analyze the sign of the numerator and denominator.

• Example 1: $\lim_{x \to 3^+} \frac{2 - 5x}{x - 3}$

• As $x \to 3^+$, numerator $2 - 5x \to -13$, denominator $x - 3 \to 0^+$

• $\frac{-13}{0^+} = -\infty$

• Example 2: $\lim_{x \to 3^-} \frac{2 - 5x}{x - 3}$

• As $x \to 3^-$, denominator $x - 3 \to 0^-$, so $\frac{-13}{0^-} = \infty$

• Example 3: $\lim_{x \to 3} \frac{2 - 5x}{x - 3}$ does not exist (DNE) since the left and right limits are not equal.

More Examples

• Example: $\lim_{x \to 3^+} \frac{x^2 - 3x + 2}{x - 3}$

• Factor numerator: $(x - 1)(x - 2)$

• As $x \to 3^+$, numerator $= 2$, denominator $= 0^+$, so $\frac{2}{0^+} = \infty$

• Example: $\lim_{x \to 5^-} \frac{-4}{(x - 5)^2}$

• Denominator $\to 0^+$, so $\frac{-4}{0^+} = -\infty$

• Example: $\lim_{x \to -5^+} \frac{x - 7}{x(x + 5)}$

• Numerator $\to -12$, denominator $\to 0^+$, so $\frac{-12}{0^+} = -\infty$

Vertical Asymptotes of Rational Functions

Finding Vertical Asymptotes

Vertical asymptotes occur at values of $x$ that make the denominator zero but not the numerator. To find vertical asymptotes for a rational function:

• Factor numerator and denominator.

• Cancel common factors (removable discontinuities).

• Set the denominator equal to zero and solve for $x$.

Example: $f(x) = \frac{x^2 - 8x + 15}{x^2 - 5x + 6}$

• Factor numerator: $(x - 5)(x - 3)$

• Factor denominator: $(x - 3)(x - 2)$

• Cancel $(x - 3)$: $f(x) = \frac{x - 5}{x - 2}$

• Vertical asymptote at $x = 2$ (since denominator is zero and numerator is not zero).

Evaluating Limits at Vertical Asymptotes

• $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} \frac{x - 5}{x - 2}$

• As $x \to 2^+$, numerator $\to -3$, denominator $\to 0^+$, so $\frac{-3}{0^+} = -\infty$

• $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} \frac{x - 5}{x - 2}$

• As $x \to 2^-$, denominator $\to 0^-$, so $\frac{-3}{0^-} = \infty$

• $\lim_{x \to 2} f(x)$ does not exist (DNE) since the left and right limits are not equal.

Summary Table: Infinite Limits and Vertical Asymptotes

Function	Point of Interest	Left Limit	Right Limit	Limit Exists?	Vertical Asymptote?
$\frac{1}{x}$	$x \to 0$	$-\infty$	$\infty$	No (DNE)	Yes ($x = 0$)
$\frac{2 - 5x}{x - 3}$	$x \to 3$	$\infty$	$-\infty$	No (DNE)	Yes ($x = 3$)
$\frac{x - 5}{x - 2}$	$x \to 2$	$\infty$	$-\infty$	No (DNE)	Yes ($x = 2$)

Key Takeaways

• Infinite limits occur when a function grows without bound as $x$ approaches a certain value.

• Vertical asymptotes are lines where the function approaches infinity or negative infinity.

• To evaluate infinite limits, analyze the sign of the numerator and denominator as $x$ approaches the point of interest.

• If the left and right limits are not equal, the limit does not exist (DNE).