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Problem 36 Medium Difficulty

Use the Squeeze Theorem to show that
$$ \lim_{x \to 0}\sqrt{x^3 + x^2}\sin \frac{\pi}{x} = 0 $$

Illustrate by graphing the functions $ f $, $ g $, and $ h $ (in the notation of the Squeeze Theorem) on the same screen.

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Video Transcript

this problem Number thirty six on the sewer calculus a fetish in section two point three. Use the squeeze. The're, um, to show that the limit is experts. Zero of the square root of the quantity X cubed plus X squared, multiplied by sine of pie over X is equal to zero. And then we will illustrate there crafts of the functions to confirm on this result. So to recall, the squeeze serum says that if the limited six approaches, A of H is equal to a limit hell and the limit is expression. A of G is also l and half is a function that is defined as greater than H critics and are equal to each less than or equal to g man. The limit is expertise. E of f is also going to tell two. We will begin to confirm and confirmed this result Bye, beginning with the fact that sign in the function there oscillates between negative one and positive one. And we will take this quantity and multiplied to the left and the right side. I think her square root with quantity execute plus X squared and at this point, as long as we can confirm that the limit as experts zero on dysfunction and dysfunction, are both the same. We can make a conclusion about this middle limit, whether it's equal to zero. So the limited six pretty zero negative square root ten minute quantity x cubed plus X squared. This is most definitely zero and the limit. It's exposure zero of the square root of the quantity X cute plus X squared also zero. So if the's tour zero and disfunction lies between those two functions them, this Lim is definitely also equal to zero. My squeeze there Now we will illustrate the functions and show exactly this notice. The brain function is always less then the orange function, which is the function of interest. The purple functions always greater then and in this area Albertine zero we see that the limits definitely too. Approach zero for all three functions