Find the limit. lim x → 2 x 2 − 7 x + 12 x − 3 \lim_{x\rarr2}\frac{x^2-7x+12}{x-3} lim x → 2 x − 3 x 2 − 7 x + 12

In this problem, we're asked to find the limit of x squared minus 7x plus 12 over x minus 3 as x approaches 2. Now your first instinct here might be to just go ahead and start factoring, but let's remember that we need to check if our denominator is 0 or not because if it's not going to be 0 then our life is a whole lot easier and we don't have to worry about factoring. Now here we're looking at x approaching 2. If I plug 2 into my denominator here I get 2 minus 3 which will end up being negative 1. Now this is not 0, which means that I can proceed in just plugging 2 in to this problem in order to get my limit. So let's go ahead and do that here. I don't have to factor anything. Now plugging in 2, I get 2 squared minus 7 times 2 plus 12 and I already figured out what that denominator is 2 minuteus 3 which is negative 1. Now doing some further algebra here 2 squared is 4, 7 times 2 is 14, so this is 4 minus 14 plus 12 all divided by negative 1. Now 4 minus 14 will give me negative 10 plus 12 will give me 2. So this is going to be 2 over negative 1 which gives me a final answer of negative 2 for the limit of this rational function as x approaches 2. So remember to double check if your denominator is actually not 0 because then your life is way easier. Thanks for watching and 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.
$\lim_{x\rarr2}\frac{x^2-7x+12}{x-3}$

$-1$

$-2$

$0$

DNE

Verified step by step guidance

First, identify the type of limit problem. This is a rational function where both the numerator and the denominator are polynomials.

Check if direct substitution of x = 2 into the function results in an indeterminate form like 0/0. Substitute x = 2 into the numerator and denominator: $ x^2 - 7x + 12 $ and $ x - 3 $.

Calculate the numerator: $ 2^2 - 7(2) + 12 = 4 - 14 + 12 = 2 $. Calculate the denominator: $ 2 - 3 = -1 $. Since the denominator is not zero, direct substitution is possible.

Since direct substitution does not result in an indeterminate form, evaluate the limit by substituting x = 2 directly into the function: $ \frac{2^2 - 7(2) + 12}{2 - 3} $.

Simplify the expression to find the limit. The limit is the value of the function as x approaches 2.