Find the following limits or state that they do not exist. Ass...

Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→1 x^2 − 1 / x − 1

Accepted Answer

Welcome back, everyone evaluate the limit or state that it does not exist limit of Z squared minus four divided by Z plus 2 1. Z approaches negative two. We're given for answer choices. A says negative one, B says negative two, C negative four and D says the limit does not exist. So essentially, let's go ahead and apply direct substitution. First of all, Z approaches negative two, we have a rational function Z squared minus four divided by Z plus two. If we apply direct substitution, we end up with negative T squared which is four, we are subtracting four. And then in the denominator, we have negative two plus two. Now this is zero divided by zero, which is an indeterminate form. So whenever we have an indeterminate form, we want to somehow simplify it and cancel out the denominator. If it's possible. In this case, if we carefully analyze the numerator, we can see that the numerator can be factored out Z squared minus four is essentially the same as Z squared minus two squared. We have a difference of squares, right. So we can apply the formula A squared minus B squared equals a minus B multiplied by A plus B. Our A is Z and our B is two because four is two squared. So we end up with C minus two, multiplied by Z plus two in the numerator. And our denominator has Z plus two. So we can cancel out Z plus two, which helps us eliminate the indeterminate form. And the limit becomes limit when Z approaches negative two of Z minus two. And now we can apply direct substitution once again limit. When Z approaches negative two of Z minus two, applying direct substitution, we get negative two minus two, which is equal to four. Now let's careful look at the answer choices, negative two minus two gives us negative four. So the correct answer choice to this problem is option C negative four. Thank you for watching.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→1 x^2 − 1 / x − 1

Key Concepts

Limits

Factoring

Indeterminate Forms

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