Study Notes: Infinite Limits in Calculus

Limits and Continuity

Infinite Limits

In calculus, infinite limits describe the behavior of a function as it increases or decreases without bound near a certain point. These limits help us understand vertical asymptotes and the unbounded growth of functions.

• Definition: Let f be a function defined on both sides of a, except possibly at a itself. We say:

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

if the values of f(x) can be made arbitrarily large (positive) by taking x sufficiently close to a (but not equal to a).

• Interpretation: This means that as x approaches a, f(x) increases without bound.

• Similarly: $\lim\limits_{x \to a} f(x) = -\infty$ means f(x) decreases without bound as x approaches a.

Examples of Infinite Limits

• Example 1: $\lim\limits_{x \to 2} \frac{1}{(x-2)^2} = \infty$

• Example 2: $\lim\limits_{x \to 0} \frac{1}{x^2} = \infty$

Both examples illustrate that as x approaches the specified value, the denominator approaches zero, causing the function to grow without bound.

General Case

• For any real number a and positive integer n:

$\lim\limits_{x \to a} \frac{1}{(x-a)^n} = \infty$

as x approaches a from either side, provided n is even.

Special Cases and Observations

• Example: $\lim\limits_{x \to 4} \frac{1}{(x-4)^2} = \infty$

• Observation: $\lim\limits_{x \to 0} \frac{\sin(x)}{x} = 1$ (This is a classic limit, not infinite, but often discussed in the context of limits approaching zero.)

Additional info: Infinite limits are closely related to the concept of vertical asymptotes in graphing functions. When a function approaches infinity as x approaches a certain value, the graph of the function will have a vertical asymptote at that value.