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How do you evaluate the limit #lim_(x to -1) (3x)/(x^2+2x+1)#?

Answer 1

The limit does not exist

The first method that should always be checked when calculating a limit is substitution. In this case, we substitute the value -1 into the expression. That yields #(-3)/0# and this value is undefined. When a value of a limit is undefined, a limit at that point does not exist.

I would like to point out that an undefined value is different from an indeterminate form. There are many indeterminate form, and the most common indeterminate form that does not include infinity is #0/0#. However, if there is some value over zero, this is different, like in the solution to this specific limit. Any value over zero is undefined and that results in a non-existent limit.

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Answer 2

Aurora Alvarado

To evaluate the limit lim_(x to -1) (3x)/(x^2+2x+1), we can substitute -1 for x in the expression. Doing so, we get (3*(-1))/((-1)^2+2*(-1)+1). Simplifying further, we have -3/(1-2+1). Continuing to simplify, we get -3/0. Since division by zero is undefined, the limit does not exist.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.