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.

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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Multiple Choice

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

$negative 9$

$negative 2$

$0$

Does not exist

Verified step by step guidance

Identify the type of limit problem: This is a rational function limit problem where both the numerator and the denominator are polynomials.

Check for direct substitution: Substitute x = -2 into the function $ \frac{x^2 - 5x - 14}{x + 2} $. If this results in an indeterminate form like $ \frac{0}{0} $, proceed to the next step.

Factor the numerator: The numerator $ x^2 - 5x - 14 $ can be factored into $ (x - 7)(x + 2) $.

Simplify the expression: Cancel the common factor $ (x + 2) $ from the numerator and the denominator, resulting in the simplified expression $ x - 7 $.

Evaluate the limit of the simplified expression: Substitute x = -2 into the simplified expression $ x - 7 $ to find the limit.