Chapter 2.2, Problem 50E

(a) Evaluate h(x) = (tan x – x)/x³ for x = 1, 0.5, 0.1 , 0.05, 0.0 1, 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 pan (b) is correct? Explain why you eventually obtained 0 values. (In Section 4.4 a method for evaluating this limit will be explained.)

(d) Graph the function h in the viewing rectangle [–1, 1] by [0, 1]. Then zoom in toward the point where the graph crosses they-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 pan (c).