Limits: Infinite Limits and Lateral Limits

Limits and Continuity

Infinite Limits

Infinite limits describe the behavior of a function as it approaches a certain point, where the function's value increases or decreases without bound. This concept is fundamental in calculus for understanding discontinuities and asymptotic behavior.

• 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.

• Example: The function $f(x) = \frac{1}{(x-2)^2}$ as $x \to 2$ approaches infinity.

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

• Example: The function $f(x) = \frac{1}{x^2}$ as $x \to 0$ approaches infinity.

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

• General Rule: For any real number a,

$\lim_{x \to a} \frac{1}{(x-a)^n} = \infty$ for even $n$

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

• Example: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ (not infinite, but a classic limit)

Lateral Limits

Lateral limits (or one-sided limits) provide additional information about the behavior of a function as it approaches a point from one side only. This is important for analyzing discontinuities and asymptotes.

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

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

Key Points:

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

• Lateral limits help distinguish between different behaviors on either side of a point.

Additional info: The notes also briefly mention the classic limit $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$, which is a fundamental result in calculus, though not an infinite limit.