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Evaluate the limit, if it exists.

$ \displaystyle \lim_{x \to -3}\frac{x^2 + 3x}{x^2 - x - 12} $

$\frac{3}{7}$

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Campbell University

Oregon State University

Harvey Mudd College

Boston College

Okay we want to find a limit of this expression as extra approaches -3. So what we're going to do first is we're going to factor the numerator and the denominator and see if we can cancel any common factors. So we're taking the limit as x approaches negative three of x squared plus three X over X squared minus x minus 12. Well, X squared plus three X. If we factor out the greatest common factor of X, X squared plus three X factors into X times X plus right, X times X is X squared X times three is three X. The denominator X squared minus X minus 12. Will factor into X minus four times X plus straight. If we use foil to double check our work X times X is X squared outer positive three X inner negative four X. They combine to give you negative one X Last Times last -4 times positive 3 -12. No, we can cancel out the common factors of X plus three. As long as X plus three is not zero because we can't have zero in the denominator can divide by zero, X plus three would only be zero. If x equal negative three. We are taking the limit as X approaches negative three. But when we take a limit as X approaches negative three, X never actually gets to equal negative three, it just approaches negative three. So X will never be negative three. And so we don't have to worry about this being zero. So we can cancel out the common factors of X plus three. Now, uh the limit of this expression X over x minus four is all ever means. So we need to take the limit of X over x minus four as X approaches negative three. Now, as x approaches -3, uh That means this acts is simply going to be approaching -3. The X -4 Term as X approaches -3, X -4 will approach -3 -4, which is -7. So as x approaches -3, the numerator is approaching negative three. The denominator is approaching negative seven. And so this quotient is approaching negative three over negative seven or positive 37 So negative divide by negative is positive, so as X approaches negative three, this expression approaches three sevens. So the limit of our function as X approaches negative three is 3/7.