Answer
$$\lim_{x \to 0}\frac{\cos x-1}{x}=0$$
$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline
x & -0.1 & -0.01 & -0.001 & 0 & 0.001 & 0.01 & 0.1 \\ \hline
f(x) & 0.04995834 & 0.00499995 & 0.0005 & \text{undefined} & -0.0005 & -0.00499995 & -0.04995834 \\ \hline
\end{array}$$
Work Step by Step
Looking at the graph, we can find that when $x$ approaches $0$ from left and right, the function approaches $0$. So we can conclude that $\lim_{x \to 0} \frac{\cos x-1}{x}$ is approximately $0$. The table also confirms our conclusion.
Now, we want to find the limit analytically.$$\lim_{x \to 0}\frac{\cos x-1}{x}=\lim_{x\to 0}- \left (\frac{1-\cos x}{x} \right )=-(0)=0$$(In the calculation of the limit we have used Theorem 1.9 (2), $\lim_{x \to 0}\frac{1-\cos x}{x}=0$).
