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Numerade Educator

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Problem 60 Medium Difficulty

If $ \displaystyle \lim_{x \to 0}\frac{f(x)}{x^2} = 5 $, find the following limits.

(a) $ \displaystyle \lim_{x \to 0}f(x) $ (b) $ \displaystyle \lim_{x \to 0}\frac{f(x)}{x} $

Answer

a. $\lim _{x \rightarrow 0} f(x)=0$
b. $\lim _{x \rightarrow 0} \frac{f(x)}{x}=0$

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Video Transcript

this is problem number sixty of the Stuart Calculus eighth edition section two point three If the limit as X approaches zero of the function F of X, divided by X squared, equals five, find the following limits. The limit is experts zero of ethics for Party and the limiters Expertise. Hero of the function f divided by X for part B for party. We take a look at the original statement and we use one of our properties and limits or potions to rewrite thiss limit as limited f right away. The limit as experts. Zero of X squared equals two five and we can solve for the limit as experts zero of the function if by not playing both sides part of the limit is that X approaches hero of the function X squared, the limit is export zero of X squared zero and then five times zero gives the answer for party and zero part of being We will take Ah, we will take this original limit and split it up a different way instead of splitting it up as one limit of after matter by the limited X squared. Well, just take one of the exes from the denominator and keep it here with that for the numerator down here, we're just up with X. This is still a quart of five. We just He was one of our limit, cause for the oceans. And then here we divide that are sorry. Here we go. Fly by The limit is experts zero of X from both sides. Leaving David, is there a limit on left? And so we have the limiters expressions hero of F over X, which is that terrible man on left equals front of tender limit of X as experts zero. And of course, this is your own, as experts zero zero times five equals zero. And so we have our answer apart to be a swan.