In this post, we’ll learn a step-by-step approach to solve the following limit problem: \(\displaystyle \lim_{x \to 0} \frac{1 – \cos(x)}{x^2} \)
If we substitute x=0 directly, we encounter a divide-by-zero problem. To solve this, we need to convert the given expression in some other form so that we do not have a divide by 0 issue. Let’s dive into the solution!
Solution:
\( \displaystyle \lim_{x \to 0} \frac{1 – \cos(x)}{x^2} \\ \displaystyle = \lim_{x \to 0} \frac{2 \sin^2\left(\frac{x}{2}\right)}{x^2} \scriptsize \text{, using the trigonometric identity} \\ \\ \small \text{let } u = \frac{x}{2}. \text{ As } x \to 0, u \to 0 \text{. so, using the substitution} \\ \\ \displaystyle = \lim_{u \to 0} \frac{2 \sin^2\left(u\right)}{(2u)^2} \\ \displaystyle = 2 \cdot \frac{1}{4} \cdot \lim_{u \to 0} \frac{\sin^2\left(u\right)}{u^2} \\ \displaystyle = 2 \cdot \frac{1}{4} \cdot \lim_{u \to 0} \left( \frac{\sin(u)}{u} \right)^2 \\ \displaystyle = \frac{1}{2} \cdot 1 \small \text{, since } \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \\ \displaystyle = \frac{1}{2} \)Please let me know in the comments if you find any error in this solution.