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Problem 29 Easy Difficulty

(a) By graphing the function $ f(x) = (\cos 2x - \cos x)/x^2 $ and zooming in toward the point where the graph crosses the $ y $ -axis , estimate the value of $ \displaystyle \lim_{x \to 0}f(x) $.

(b) Check your answer in part (a) by evaluating $ f(x) $ for values of $ x $ that approach 0.

Answer

$$\text { (a) From the graphs, it seems that } \lim _{x \rightarrow 0} \frac{\cos 2 x-\cos x}{x^{2}}=-1.5$$
(b) $$\begin{array}{|l|l|}
\hline x & f(x) \\
\hline\pm 0.1 & -1.493759 \\
\pm 0.01 & -1.499938 \\
\pm 0.001 & -1.499999 \\
\pm 0.0001 & -1.500000 \\
\hline
\end{array}$$

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Video Transcript

this is problem number twenty nine Stuart Calculus eighth edition. Section two point two Party By grafting the function at FedEx is equal to the quantity of co sign two. X Man is cosign of eggs divided by X squared and zooming in towards the point where the graph crosses the Y axis has to meet. The value of the limit is X approaches. Zero for the function X. We're going to take this function cosign of two x minus cosign of X. That whole quantity, the better. Right X squared. And we're going to use a graphing calculator or, in his case, a Griffin spreadsheet. And we're going to investigate what the behavior of the function is as we approach X equals zero from the left and from a rating. So here's a plot of the function. We've already zoomed in very closely. Two. The area of X equals zero, which is this line as we a closer from the left, the function of purchase a value of about negative one point five. As we approach from the right, we also purchase value that's close to negative one point five. And so our first estimate for party Yeah, that we come from graphically is that the limit is equal to negative one point five. We're gonna check this answer and party by evaluating the function for values of X that approach zero. And if we go back to a spreadsheet, you see that this plot was actually made using values that we're very close to zero and we will focus specifically on the table of values and show that from the right hand side as we hear closer and closer, zero from the right, we see that the values got closer and closer to negative one point five Similarly, from the left values that are negative and close to zero, we see the calculations getting closer and closer to about negative one point five Toy reached this point. At which point we can say defendant Arlene, that power lim as explosions Zero for this function is negative. One point five and we have confirmed too pretty