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Chapter 1: Problem 49

Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 0} \frac{\sin x^{2}}{x} $$

Short Answer

Expert verified
The limit as \(x\) approaches 0 of the function \(\frac{\sin x^{2}}{x}\) is 0.

Step by step solution

01

Graph and Estimate the Limit

First, graph the function \(\frac{\sin x^{2}}{x}\) using a graphing utility. Look closely around \(x = 0\). The graph should show that as \(x\) approaches zero, the value of the function also approaches zero, so our estimated limit is 0.
02

Use a Table

Make a table of values close to 0. The table should have two columns - one for \(x\) values and the other for \(f(x)\) values where \(f(x)\) = \(\frac{\sin x^{2}}{x}\). Fill in the table with \(x\) values approaching 0 (values like 0.1, 0.01, 0.001, and their negative counterparts). Compute the corresponding \(f(x)\) values. The table should show that as \(x\) approaches 0, so does \(f(x)\), further corroborating our initial estimate of 0.
03

Analytical Method

For \(x\) approaching 0, the given function is in the indeterminate form \(0/0\). Here, we apply L'Hopital's Rule, which states that: If the limit of a function as \(x\) approaches a certain value results in the form \(0/0\) or \(\infty/\infty\), then this limit can be obtained by taking the limit of the derivative of the numerator divided by the derivative of the denominator around that \(x\) value. Constructs the derivatives: \(f'(x)\) = \(2x\cos(x^2)\) and \(g'(x)\) = 1. Evaluate the limit using L'Hopital's rule: \(\lim_{x \rightarrow 0} \frac{f'(x)}{g'(x)}\). Apply the limit to the derivatives which gives: \(\lim_{x \rightarrow 0} 2x\cos(x^2) = 0\).
04

Conclusion

The results obtained from all three methods, graphing, table and analytical, have been consistent. Hence, it concludes that the limit of the function \(\frac{\sin x^{2}}{x}\) as \(x\) approaches 0, is indeed 0.

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Most popular questions from this chapter

Use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ h(\theta)=1+\theta-3 \tan \theta $$

Prove that if \(\lim _{\Delta x \rightarrow 0} f(c+\Delta x)=f(c),\) then \(f\) is continuous at \(c\)

Verify each identity (a) \(\arcsin (-x)=-\arcsin x, \quad|x| \leq 1\) (b) \(\arccos (-x)=\pi-\arccos x, \quad|x| \leq 1\)

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In the context of finding limits, discuss what is meant by two functions that agree at all but one point.
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