Sketch graphs by hand and use them to make approximations...

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow 1} \frac{x-1}{x^{2}-1}
$$

Key Concepts

Limit

A limit in calculus represents the value that a function approaches as its input approaches some value. It captures the behavior of the function near a point, even if the function is not defined exactly at that point. This concept is fundamental for understanding continuity and the behavior of functions in calculus.

Removable Discontinuity

A removable discontinuity occurs when a function has an undefined point or a hole, but the limit exists because the other values of the function near that point behave consistently. This is often seen in rational functions where common factors in the numerator and denominator cancel out, allowing the limit to be determined even if the function is not defined at a specific input.

Rational Function Simplification

Rational functions are ratios of polynomials that often can be simplified by factoring and canceling common factors. This simplification can reveal the underlying behavior of the function and is particularly useful when evaluating limits, as it may eliminate indeterminate forms such as 0/0.

Graphical Analysis

Graphical analysis involves sketching or plotting a function to visually inspect its behavior near points of interest. This technique helps in approximating limits by showing how the function behaves as it approaches a certain value, and it can also illustrate the existence of discontinuities or asymptotes.

One-Sided Limits

One-sided limits analyze the behavior of a function as the input approaches a point from one direction only (either from the left or from the right). They become especially important when a function does not have a common limit from both sides, leading to different behaviors that may require separate analysis.

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