Infinite Limits and Vertical Asymptotes

Infinite Limits

Definition and Concept

An infinite limit occurs when the values of a function increase or decrease without bound as the input approaches a certain value. This typically happens near points where the function is undefined, such as where the denominator of a rational function is zero.

• If the function values grow arbitrarily large in the positive direction as x approaches a value a from both sides, we write:

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

• If the function values grow arbitrarily large in the negative direction as x approaches a from both sides, we write:

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

In both cases, the limit does not exist in the conventional sense.

One-Sided and Two-Sided Infinite Limits

• Infinite limits can be one-sided (approaching from only the left or right) or two-sided (approaching from both sides).

• They are often associated with vertical asymptotes of the function.

Vertical Asymptotes

Definition and Occurrence

• A vertical asymptote occurs at x = a if the function increases or decreases without bound as x approaches a.

• For rational functions, vertical asymptotes occur at x-values that make the denominator zero, provided the factor does not cancel with the numerator (which would instead create a hole).

• Vertical asymptotes also occur in transcendental functions such as the tangent, cotangent, and logarithmic functions.

Examples: Finding Vertical Asymptotes

• Example 1: $f(x) = \frac{x^2 - 9x + 14}{x^2 - 5x + 6}$ Factor numerator and denominator: $f(x) = \frac{(x-7)(x-2)}{(x-3)(x-2)}$ Hole at $x=2$ (factor cancels), vertical asymptote at $x=3$.

• Example 2: $f(x) = \frac{x-5}{x^2-25}$ $f(x) = \frac{x-5}{(x+5)(x-5)}$ Hole at $x=5$, vertical asymptote at $x=-5$.

• Example 3: $f(x) = \frac{x^2-3x+2}{x^{10}-x^9}$ $f(x) = \frac{(x-2)(x-1)}{x^9(x-1)}$ Hole at $x=1$, vertical asymptote at $x=0$.

• Example 4 (Transcendental): $g(\theta) = \tan\left(\frac{\pi \theta}{10}\right)$ Vertical asymptotes where $\cos\left(\frac{\pi \theta}{10}\right) = 0$ $\frac{\pi \theta}{10} = \frac{\pi}{2} + \pi k$ $\theta = 5 + 10k$, $k \in \mathbb{Z}$

Analyzing Infinite Limits from Graphs

Graphical Interpretation

• When the graph of $f(x)$ approaches $+\infty$ or $-\infty$ as $x$ approaches a certain value, there is a vertical asymptote at that value.

• If the left- and right-hand limits are both $+\infty$ or both $-\infty$, the two-sided limit is infinite (does not exist in the conventional sense).

• If the left- and right-hand limits are different (one $+\infty$, one $-\infty$), the two-sided limit does not exist.

Example Table: Interpreting Limits from Graphs

Limit Expression	Graphical Behavior	Result
$\lim_{x \to 1^-} f(x)$	Approaches $+\infty$	$+\infty$
$\lim_{x \to 1^+} f(x)$	Approaches $+\infty$	$+\infty$
$\lim_{x \to 1} f(x)$	Both sides $+\infty$	$+\infty$
$\lim_{x \to -3^-} f(x)$	Approaches $+\infty$	$+\infty$
$\lim_{x \to -3^+} f(x)$	Approaches $-\infty$	$-\infty$
$\lim_{x \to -3} f(x)$	Different one-sided limits	DNE

Determining Infinite Limits Algebraically

Key Steps

• Factor and simplify the function if possible.

• Identify points where the denominator is zero (potential vertical asymptotes).

• Check if the numerator is also zero at those points (if so, there may be a hole instead).

• Analyze the sign of the numerator and denominator as x approaches the critical value from the left and right.

Example Calculations

• $\lim_{x \to 4} \frac{x-5}{(x-4)^2}$ As $x \to 4$, numerator $\to -1$, denominator $\to 0^+$, so limit $= -\infty$.

• $\lim_{x \to -2} \frac{x-4}{x^2+2x}$ As $x \to -2$, numerator $\to -6$, denominator $\to 0^+$, so limit $= -\infty$.

• $\lim_{x \to 2} \frac{1}{\sqrt{x(x-2)}}$ As $x \to 2^+$, denominator $\to 0^+$, so limit $= +\infty$; from the left, the expression is not defined (DNE).

• $\lim_{x \to 0^+} \csc x = \lim_{x \to 0^+} \frac{1}{\sin x}$ As $x \to 0^+$, $\sin x \to 0^+$, so limit $= +\infty$.

Summary Table: Infinite Limits and Vertical Asymptotes

Function Type	Asymptote Location	Behavior
Rational	Zeros of denominator (not canceled)	Infinite limit (vertical asymptote)
Transcendental (e.g., tan, log)	Zeros of denominator or argument	Infinite limit (vertical asymptote)
Hole (removable discontinuity)	Zero canceled in numerator and denominator	No vertical asymptote

Key Takeaways:

• Infinite limits indicate unbounded growth near a point and are associated with vertical asymptotes.

• Careful algebraic and graphical analysis is required to distinguish between vertical asymptotes and holes.

• One-sided limits help determine the precise behavior near points of discontinuity.