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Problem 65 Hard Difficulty

(a) Use the Squeeze Theorem to evaluate $ \displaystyle \lim_{x \to \infty} \frac{\sin x}{x} $.
(b) Graph $ f(x) = (\sin x)/x $. How many times does the graph cross the asymptote?

Answer

a. $\lim _{x \rightarrow \infty} \frac{\sin x}{x}=0$
b. Thus, the graph crosses the asymptote an infinite number of times.

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Video Transcript

This is problem number sixty five of this tour Calculus eighth edition, Section two point six. Use the squeeze staring to evaluate the Limited's expertise in penalty. A sign of X over Now are squeezed there, Miss Cui recall is if we have two three functions age Afrin, G and F is a function that is greater than or equal to h but less than or equal to G and the limit as experts. Zane, some value ain of age is equal to the limit. Is expert is eight for the function gene, and this is they're both equal to each other to some limit at all. Therefore, for these reasons, the result is that the limit is expert is a for the function have it is also equal down and that is the definition of are squeezed here. Um, we're going do this for our problem here. Com. We know that sign of X is a function that arse awaits between the values negative or a positive one. If we divide each turn by X, we get negative. Wanna Rex less than equal to sign X or ex less than equal to one a rex. And now, if we just take the limit. Ah, Of each term as ex president, well has expressions impunity for negative One of rags approaches zero since a very large number of the denominator it means of this number increasingly grows smaller. So the limit hero be zero. The same reasoning goes to proving that the limit its export is infinity. One over hex goes through. So that must mean they're because the planet is expressive unity of negative Wanna wreck zero And the Limited Express in India won a rec zero. Then the limited Express infinity of sand over Cenex Rex must also be zero. All right, preppie graph vexes sine x rex. How many times is a graft crosses the asked me to So when it talks about the Assam took, I recall that we define horizontal ass and tits as the limit and six persons infinity of a certain function. So technically, here we have proven that why equal zero is a horizontal as him total dysfunction on. So, by looking at the ground, we identify the horizontal awesome totals the X axis and we see that this function crosses thie XX is an infinite amount of time. Many, many times are oscillating up and down over the X axis, and the function approaches zero as expressions infinity but isn't exactly equal to zero. So the answers and an infinite amount of times.