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Numerade Educator

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Problem 30 Medium Difficulty

Evaluate the limit, if it exists.

$ \displaystyle \lim_{x \to -4}\frac{\sqrt{x^2 + 9} - 5}{x + 4} $

Answer

$-.8$

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Video Transcript

this problem Number thirty of the Stuart Calculus eighth edition section two point three Evaluate the limit If it exists the limit as X approaches Negative for the square root of the quantity X squared plus line minus five Divided by the quantity X plus. For with problems like these, it helps to rationalize the numerator and multiply. We do this, but I'm on the plane. Bright d congregate. In this case, it's the square root of the quantity X scripless time plus five to the top and the bottom. If we do this, the numerator simplifies to X squared plus nine minus twenty five. And then the denominator is the quantity X was for one supplied by the quantity of the screw it of the quantity export last time plus fire. If we simplify the animator, this becomes negative sixteen and we see that the numerator can be factored because it is a difference of squares. It can be factored into X plus four X wings for and the denominator will stay the same. So we have rationalized the numerator we have eliminated or we plan to eliminate this X plus four turn. And now we have a simplified I'm in. So experts is negative for of X minus horror directed by square root of the quantity X squared plus nine plus fire. And as we've only way, we get negative for my pants for and the narrator here the denominator we have the square root of I need a four squared sixteen plus nine plus five sixteen plus minus twenty five. The square root of twenty five is time. So we have negative on the numerator over a pipe last time, which is ten. And this could be simplified into our final answer of negative for over five.