Limits and Infinite Limits: Study Notes

Limits and Continuity

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a particular 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 we can make the values of f(x) arbitrarily large (or small) by taking x sufficiently close to a, but not equal to a.

• Key Point: Infinite limits describe the behavior of functions near points where they diverge to positive or negative infinity.

• Example: The function $f(x) = \frac{1}{x}$ as $x \to 0$ diverges to infinity or negative infinity depending on the direction of approach.

One-Sided Limits

One-sided limits analyze the behavior of a function as the input approaches a point from one side only (left or right). This is useful for functions that behave differently on either side of a point.

• Left-hand limit: $\lim_{x \to 0^-} \frac{1}{x} = -\infty$

• Right-hand limit: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$

• Application: These limits help describe discontinuities and asymptotic behavior.

Examples of Infinite Limits

Several classic examples illustrate infinite limits:

• 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: $\lim_{x \to a} \frac{1}{(x-a)^n} = +\infty$ for all even integers n.

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

• Special Case: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ (not infinite, but important for comparison).

Additional info: Infinite limits are closely related to vertical asymptotes, where the function grows without bound near a specific value of x.