Question

Estimate the limit. Picture Below

Answer 1

Answer:

Hence, the limit of the expression $\lim_{x \to 3} \dfrac{x-3}{x^2-9}$ is:

0.1667 (i.e. option b is true)

Step-by-step explanation:

We are asked to estimate the limit of the expression:

$\lim_{x \to 3} \dfrac{x-3}{x^2-9}$

We know that:

$a^2-b^2=(a-b)(a+b)$

Hence, we could represent it as:

$\lim_{x \to 3} \dfrac{x-3}{x^2-3^2}\\ \\= \lim_{x \to 3} \dfrac{x-3}{(x-3)(x+3)}\\ \\= \lim_{x \to 3} \dfrac{1}{x+3}$

Since we cancel out the similar terms in the numerator as well as in the denominator.

$\lim_{x \to 3} \dfrac{1}{x+3}=\dfrac{1}{3+3}=\dfrac{1}{6}=0.1667$

Hence, the limit of the expression $\lim_{x \to 3} \dfrac{x-3}{x^2-9}$ is:

0.1667

Answer 2

Answer:

Choice b is the answer.

Step-by-step explanation:

We have given a function.

f(x) = x-3 / x²-9

We have to find the limit of given function at x.

$\lim_{x \to \ 3} x-3/x^{2} -9$

Applying difference formula to denominator of given function.

x-3 / x²-9 = x-3 / (x-3)(x+3)

x-3 / x²-9 =  1 / x+3

Applying limit, we have

$\lim_{x \to \ 3} x-3/x^{2} -9$  =  $\lim_{x \to \ 3} 1/x+3$

$\lim_{x \to \ 3} x-3/x^{2} -9$  = 1/3+3

$\lim_{x \to \ 3} x-3/x^{2} -9$  = 1/6

$\lim_{x \to \ 3} x-3/x^{2} -9$  = 0.1667 which is the answer.