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

Business Calculus

1. Limits and Continuity

Finding Limits Algebraically

Struggling with Business Calculus?

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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

Step 1: Recognize that the given problem involves finding the limit of a rational function as x approaches -2. The function is $ \frac{x^2 - 5x - 14}{x + 2} $. Start by checking if direct substitution of $ x = -2 $ into the function results in an indeterminate form (e.g., $ \frac{0}{0} $).

Step 2: Substitute $ x = -2 $ into the numerator $ x^2 - 5x - 14 $ and denominator $ x + 2 $. For the numerator, calculate $ (-2)^2 - 5(-2) - 14 $. For the denominator, calculate $ -2 + 2 $. If both numerator and denominator equal zero, the expression is indeterminate, and further simplification is needed.

Step 3: Factorize the numerator $ x^2 - 5x - 14 $. Look for two numbers that multiply to $ -14 $ and add to $ -5 $. The factorization is $ (x - 7)(x + 2) $. Rewrite the function as $ \frac{(x - 7)(x + 2)}{x + 2} $.

Step 4: Simplify the expression by canceling the common factor $ x + 2 $ in the numerator and denominator. This is valid as long as $ x \neq -2 $, since division by zero is undefined. The simplified function is $ x - 7 $.

Step 5: Evaluate the limit of the simplified function $ x - 7 $ as $ x \to -2 $. Substitute $ x = -2 $ into $ x - 7 $ to find the limit. The result is the value of the limit.