Use the graph to determine each limit (if it exists). Then...

Use the graph to determine each limit (if it exists). Then identify another function that agrees with the given function at all but one point.
$f(x)=\frac{x^{2}-1}{x+1}$
(a) $\lim _{x \rightarrow 1} f(x)$
(b) $\lim _{x \rightarrow 2} f(x)$
(c) $\lim _{x \rightarrow-1} f(x)$

Key Concepts

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. They allow us to analyze functions at points where the function might not be explicitly defined or where its behavior may differ from its value at that point.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in a simpler form by factoring and canceling common factors. In the context of rational functions, this technique can eliminate indeterminate forms by reducing the expression to its simplest equivalent, provided that any cancellation does not ignore essential restrictions on the domain.

Removable Discontinuity

A removable discontinuity occurs when a function is not defined at a certain point due to a factor that appears in both the numerator and the denominator, which can be canceled algebraically. This leads to a 'hole' in the graph at that point, even though the limit exists, as the simplified form of the function can be used to determine the behavior near that point.

Alternate Function Definition

An alternate function definition refers to defining a new function that coincides with the original function everywhere except possibly at the point of discontinuity. This new function often 'repairs' the removable discontinuity by assigning the value of the limit at that point, thereby turning the function into one that is continuous over the entire domain.

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