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 (2x − 1)^2 − 9 / x + 1

Accepted Answer

Welcome back, everyone determine the limit below or state that it does not exist limit of ty minus V squared minus 49 divided by Y plus T one. Y approaches negative two. Where given, for answer choices, A says the limit does not exist B negative seven C negative 14 and D negative 28. So first of all, let's apply direct substitution. Y approaches negative two. We are evaluating the limit of a rational function to Y minus V squared minus 49 divided by Y plus two. Applying direct substitution. If Y approaches negative two, then we get two multiplied by negative two, which is negative four minus three squared minus 49 divided by negative two plus two. When we simplify that we end up with an expression of zero divided by zero, which is an indeterminate form, meaning we want to somehow deal with this indeterminate form. And we can apply factorization. Notice that in the numerator we have a difference of squares because 49 is seven squared. So we can rewrite our limit as limit when Y approach is negative two. And now the difference of squares can be factored out as a squared minus B squared equals A minus B multiplied by A plus B. Our A is two, Y minus three. Our B is seven. So we end up with two Y minus three minus seven, multiplied by two, Y minus three plus seven. In the denominator, nothing changes. We have wipe list too. So now let's simplify. We get two Y minus 10 and we can factor out the two. So we get Y minus five, multiplied by two, right? Two, Y minus 10, 2, Y minus 10. So we have factored out the two. And then for the other factor, we have two, Y minus three plus seven. So that's two Y plus four or basically Y plus two multiplied by two. Because when we distribute, we get ty plus four, ty plus four in the denominator week at Y plus two. And we can clearly see that the Y plus two factor can be canceled out. Let's simplify further. We get a limit when Y approaches negative two, two, multiplied by two, gives us four. And then what is multiplied by Y minus five? Once again, let's try direct substitution or multiplied by negative two minus five. That's four multiplied by negative seven, which is negative 28. Now this is a finite number and we can see that the correct answer is option D negative 28. 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 (2x − 1)^2 − 9 / x + 1

Key Concepts

Limits

Factoring and Simplifying Expressions

Indeterminate Forms

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