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. $$ \lim _{x \rightarrow-1} \frac{x^{3}+1}{x+1} $$

Short answer

The limit of the function as \(x \rightarrow-1\) is 3 and the simpler function that agrees with the given function at all points except x=-1 is \(x^2-x+1\).

Step-by-step solution

Function Simplification

Begin by factoring the numerator. The factorization of \(x^3+1\) is \((x+1)(x^2-x+1)\). So the function \(\frac{x^{3}+1}{x+1}\) becomes \((x+1)(x^2-x+1)\div(x+1)\).

Cancel out Common Factors

Here, \(x+1\) is a common factor in the numerator and the denominator. So we can cancel out this common factor. The function thus becomes \(x^2-x+1\). Note that our extracted simpler function is valid for all \(x\) except -1, which aligns with the exercise requirement.

Evaluate Limit

We can now evaluate the limit of the simplified function as \(x \rightarrow -1\). Substituting -1 into the simplified function \(x^2-x+1\), we get \((-1)^2-(-1)+1 = 1+1+1 = 3\). So, \(\lim _{x \rightarrow-1} \frac{x^{3}+1}{x+1}\) equals 3