In this post, we’ll learn a step-by-step approach to solve the following limit problem: \( \displaystyle \lim_{x \to 0} \frac{e^x – 1 – x}{x^2} \).
At x = 0, the given expression \( \displaystyle \frac{e^x – 1 – x}{x^2} \) is \( \frac{0}{0} \). So, we can apply L’Hôpital’s Rule to solve this limit problem.
Solution:
\( \displaystyle \lim_{x \to 0} \frac{e^x – 1 – x}{x^2} \\ \displaystyle = \lim_{x \to 0} \frac{e^x – 1}{2x} \scriptsize \text{, applying L’Hôpital’s Rule} \\ \displaystyle = \lim_{x \to 0} \frac{e^x}{2} \scriptsize \text{, applying L’Hôpital’s Rule again as expression is still } \frac{0}{0} \\ \displaystyle = \frac{e^0}{2} \\ \displaystyle = \frac{1}{2} \\ \\ \displaystyle \text{Therefore, } \fbox{$\displaystyle \lim_{x \to 0} \frac{e^x – 1 – x}{x^2} = \frac{1}{2} $} \)Please let me know in the comments if you find any error in this solution.