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.

1. Limits and Continuity

Finding Limits Algebraically

Business Calculus

1. Limits and Continuity

Finding Limits Algebraically

Struggling with Business Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

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

$-1$

$-2$

$0$

DNE

Verified step by step guidance

Step 1: Recognize that the given limit is in the indeterminate form 0/0 when directly substituting x = 2 into the function. This means we need to simplify the expression to evaluate the limit.

Step 2: Factorize the numerator $x^2 - 7x + 12$. Look for two numbers that multiply to 12 and add to -7. The factorization is $(x - 3)(x - 4)$.

Step 3: Rewrite the expression as $\frac{(x - 3)(x - 4)}{x - 3}$. Notice that $x - 3$ appears in both the numerator and denominator.

Step 4: Cancel out the common factor $x - 3$ from the numerator and denominator, leaving $x - 4$. Note that this simplification is valid only for $x \neq 3$.

Step 5: Substitute $x = 2$ into the simplified expression $x - 4$ to evaluate the limit. The result is $2 - 4$.