Chapter 1.5, Problem 44E

(a) Evaluate $h\left(x\right)=\left(\tan x-x\right)/x^3$ for $x=1,0.5,0.1,0.05,0.01,$ and $0.005$.

(b) Guess the value of $\underset{x\to 0}{\lim }\frac{\tan x-x}{x^3}$.

(c) Evaluate h(x) for successively smaller values of x until you finally reach a value of 0 for h(x). Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained values of 0 for h(x). (In Section 6.8 a method for evaluating this limit will be explained.)

(d) Graph the function h in the viewing rectangle $\left[-1,1\right]$ by $\left[0,1\right]$. Then zoom in toward the point where the graph crosses the y-axis to estimate the limit of h(x) as x approaches 0. Continue to zoom in until you observe distortions in the graph of h. Compare with the results of part (c).