Question

Find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow-1} \frac{x^{2}-1}{x+1} $$

Short answer

The limit of the function as x approaches -1 is -2. The function that agrees with the original is \(f(x) = x - 1\).

Step-by-step solution

Simplifying the function

Factorize the numerator into \((x-1)(x+1)\). This leads to a new expression of the function \(f(x) = \frac{(x-1)(x+1)}{x+1}\).

Eliminate common factors

Notice that \(x+1\) is a common factor in the numerator and the denominator, which we can cancel out. This results in a simpler function: \(f(x) = x-1\).

Calculate limit

Substitute \(x = -1\) into the simplified function to obtain the limit. This gives \(\lim _{x \rightarrow-1} (x-1) = -1 -1 = -2\).

Confirm result using a graphing utility

A graphing utility can be used to graph the function \(f(x) = x-1\) and the original function \(f(x) = \frac{x^{2}-1}{x+1}\) tro validate the result. It can be observed that the two functions agree at all points, except at \(x = -1\), where the original function is undefined.