Limits and Infinite Limits in Calculus

Limits and Continuity

Infinite Limits

Infinite limits occur when the values of a function increase or decrease without bound as the input approaches a certain point. This concept is fundamental in calculus, especially when analyzing the behavior of functions near points of discontinuity or singularity.

• Definition: Let f be a function defined on both sides of a, except possibly at a itself. We say that the limit of f(x) as x approaches a is infinite if:

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

• This means that the values of f(x) can be made arbitrarily large (or small) by taking x sufficiently close to a, but not equal to a.

• Left and Right Limits: Sometimes, we consider the behavior of the function as x approaches a from the left or right:

$\lim_{x \to a^-} f(x)$ (left-hand limit) $\lim_{x \to a^+} f(x)$ (right-hand limit)

• For example, $\lim_{x \to 0^-} \frac{1}{x} = -\infty$ and $\lim_{x \to 0^+} \frac{1}{x} = \infty$.

Examples of Infinite Limits

Infinite limits often arise in rational functions where the denominator approaches zero.

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

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

• General Case: For any real number a, $\lim_{x \to a} \frac{1}{(x-a)^n} = \infty$ for even n.

• Example 3: $\lim_{x \to 4} \frac{1}{(x-4)^2} = \infty$

• Oscillatory Example: $\lim_{x \to 0} \frac{\sin(x)}{x}$ (this limit does not tend to infinity, but is a classic example of a limit involving a singularity).

Key Points:

• Infinite limits indicate vertical asymptotes in the graph of a function.

• They are essential for understanding discontinuities and the behavior of functions near undefined points.

• Both left and right limits must be considered to fully describe the behavior at a point.

Additional info: The notes also briefly mention lateral limits and provide graphical representations to reinforce the concept of infinite limits.