Find the limit. lim x → − 2 x 2 − 5 x − 14 x + 2 \lim_{x\rarr-2}\frac{x^2-5x-14}{x+2} lim x → − 2 x + 2 x 2 − 5 x − 14

Here we want to find the limit of x squared minus 5x minus 14 over x+2 as x approaches negative 2. Now since this is a rational function, I wanna look at what's happening with my denominator. If I were to go ahead and just plug negative 2 into my function, in my denominator I would have negative 2 +2. Now because this ends up giving me 0 that means that I need to go ahead and factor here before I just plug this negative 2 in. So let's do that here. Factor here before I just plug this negative 2 in. So let's do that here. Now I'm still finding the limit as x approaches negative 2, but I want to go ahead and factor that numerator, x squared minus 5x minus 14. Now this will factor to x minus 7 times x plus 2. Thinking about two numbers that multiply to negative 14 and add to negative 5. So I have negative 7 and positive 2. Now I don't need to worry about factoring that denominator because I can't factor any further it's just x plus 2 and here I see that common factor x+2 that will end up canceling out. Now I'm just left to find the limit as x approaches negative 2 of x minus 7, which I can just plug a negative 2 into my function now. So I end up with negative 2 minus 7, which gives me a final answer of negative 9 for the limit of this function as x approaches negative 2. Let me know if you have any questions. I'll see you in the next one.

22. Limits & Continuity

Finding Limits Algebraically

Precalculus

NEW 22. Limits & Continuity

Finding Limits Algebraically

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Multiple Choice

Find the limit.
$\lim_{x\rarr-2}\frac{x^2-5x-14}{x+2}$

$-9$

$-2$

$0$

Does not exist

Verified step by step guidance

Identify the limit expression: $ \lim_{x \to -2} \frac{x^2 - 5x - 14}{x + 2} $.

Notice that direct substitution of $ x = -2 $ in the denominator $ x + 2 $ results in zero, indicating a potential indeterminate form.

Factor the numerator $ x^2 - 5x - 14 $. Look for two numbers that multiply to $-14$ and add to $-5$. These numbers are $-7$ and $2$, so the factorization is $(x - 7)(x + 2)$.

Rewrite the limit expression using the factorization: $ \lim_{x \to -2} \frac{(x - 7)(x + 2)}{x + 2} $.

Cancel the common factor $ x + 2 $ from the numerator and the denominator, resulting in $ \lim_{x \to -2} (x - 7) $. Now, substitute $ x = -2 $ to find the limit.