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 certain point. This concept is fundamental in calculus, especially when analyzing 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) as x gets 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.

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 Rule: $\lim_{x \to a} \frac{1}{(x-a)^n} = \infty$ for all n in the set of positive integers.

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

• Example 4: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ (This is a classic limit, not infinite, but often discussed in the context of limits approaching zero.)

Additional info: The notes also mention lateral limits (left and right limits), which are important when the function behaves differently from each side of the point.

• Lateral Limits: $\lim_{x \to 0^-} \frac{1}{x} = -\infty$ and $\lim_{x \to 0^+} \frac{1}{x} = \infty$

Understanding infinite limits is crucial for analyzing asymptotic behavior and discontinuities in functions.