In this post, we’ll learn a step-by-step approach to solve the following limit problem: \(\displaystyle \lim_{x \to 1} \frac{x^3 – 1}{x^2 – 1} \)
If we substitute x=1 directly, we encounter a divide-by-zero problem. To solve this, we’ll first factor the numerator and denominator. Let’s dive into the solution!
Solution:
\( \displaystyle \lim_{x \to 1} \frac{x^3 – 1}{x^2 – 1} \\ \displaystyle = \lim_{x \to 1} \frac{(x – 1)(x^2 + x + 1)}{(x – 1)(x + 1)} \scriptsize \text{ , factor the numerator and denominator} \\ \displaystyle = \lim_{x \to 1} \frac{x^2 + x + 1}{x + 1} \small \text{, cancel } (x−1) \\ \displaystyle = \frac{1^2 + 1 + 1}{1 + 1} \\ \displaystyle = \frac{3}{2} \)Please let me know in the comments if you find any error in this solution.