Problem 62 Hard Difficulty

Show by means of an example that $ \displaystyle \lim_{x \to a}\left[ f(x) + g(x) \right] $ may exist even though neither $ \displaystyle \lim_{x \to a}f(x) $ nor $ \displaystyle \lim_{x \to a}g(x) $ exists.

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Video Transcript

This is Problem number sixty two of the Stuart Calculus eighth edition section two point three show by means of an example that the limit is expert is a ah, the quantity f of X plus. DMX may exist, even though neither the limiters expert today over half nor ability as expert city of G exists now. One way to fear this example out I want to choose of inappropriate example is to choose two functions that are potentially opposites around that they themselves haven't issue. Um, with respect to Element. At a certain point, let's take after Max hands. We have a civilian ex rx for as an example. From what we know for this function, his ex approaches zero from the right. Never. Okay, get us. We get that it is X rex since approaching zero from the right mate. That's always positives that absolutely signed going, and this employs toe worn as the limit. As we approach a tear from the left, we're dealing with negative values, and so we must consider only the negative part of the function of that. So I mix, meaning that the limit is negative one. So this is an example of efforts and example of function as we approach a equals zero from less than the rate we get two different values, meaning that is limited, not exist as you approach zero. If we chose a function opposite of this a negative times a square root for the absolute value of X over X, we see that we were to the same steps show two different to different limits one negative one one as equal to one concluding that the limit as experts zero does not exist again. The same steps applying because the absolute value function changes as you're approaching zero from one end or the other Fergie of X As you approach zero from the right, your, uh, limited equal to neither one and X approaches Here on the left, you're limited approaches of one. And that means that the limit is experts Nero. For this function, gene is not exist. So if we now consider the limit as experts, zero of some of these two functions absolutely of X. Come on, Rex. Plus negative. Absolutely. Vicks Rex, We see that this clever choice of functions give this Parliament is expert zero one function plus the upset of the other. Which is it? Quit of your own. Which means it is, um it does exist, and it is equal to zero.