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Evaluate $\lim _{x \rightarrow 16} \frac{\sqrt[4]{x}-2}{x-16}$.

Calculus 1 / AB

Chapter 2

Limits

Section 3

Techniques for Computing Limits

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here. We're going to have to get a little tricky with the way that we simplify this function. So we're looking at the limit as Ex Purchase 16 of the fourth root of X minus two, divided by X minus 16. So, normally we would rationalize the numerator or wherever a radical is so we'll go ahead and do that. Well, multiply this by the fourth root of X and then a plus two. And of course, we have to do the same to the denominator. Go ahead and do that. This equals the limit as X approaches 16 432 Vex times 1/4 root of X is the square root of X. These terms should cancel. And then we're left with a minus four because positive, two times negative, too. And then the denominator gets a little messy and we'll go ahead and leave it as it is. We won't multiply that out. We have X minus 16 times 1/4 root of X plus two. And then again, we still have this radical here. So what we're gonna do is we're actually going to rationalize it one more time. So multiplying by the square root of X Plus four and again do the same to the denominator. We'll go down here and we'll say the limit as exit purchase 16. The numerator becomes brood X times your index, which is just X minus four times plus four, gives us minus 16 and then the denominator gets a little messy. But again, we won't multiply it out. X minus 16 before through decks plus two that stays. And then we're also multiplying by the square root of X plus four. So that's our entire denominator. And then finally we can cancel, and then we're left with the limit as exit purchase 16 of one over the four through of X Plus two. We're multiplying that by the square root of X Plus four, and finally, we can go ahead and plug in. Our value will go ahead and do that. We have one over the four through of 16 which is to plus two other parentheses. E. We have the square root of 16 which is four, so we have four plus four. This gives us 1/2 plus two is four. Four plus four is eight, so four times eight, which is 32 has your final answer

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