Question

Estimate the limit.
Picture Below

Answer 1

Hence, the limit of the expression
`\lim_(x \to 3) (x-3)/(x^2-9)` is:

0.1667 (i.e. option b is true)

Explanation:

We are asked to estimate the limit of the expression:

`\lim_(x \to 3) (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) (x-3)/(x^2-3^2)\\ \\= \lim_(x \to 3) (x-3)/((x-3)(x+3))\\ \\= \lim_(x \to 3) (1)/(x+3)`

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

`\lim_(x \to 3) (1)/(x+3)=(1)/(3+3)=(1)/(6)=0.1667`

Hence, the limit of the expression
`\lim_(x \to 3) (x-3)/(x^2-9)` is:

0.1667

Answer 2

Choice b is the answer.

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.