Question

Estimate the limit.
Picture below

Answer 1

Hence, the limit of the expression:

`\lim_(x \to 3) (x-3)/(x^2-2x-3)` is:

`(1)/(4)`

Explanation:

We have to estimate the limit of:

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

We can also represent the denominator of the function in the limit as:

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

Hence, we have to estimate the limit of:

`\lim_(x \to 3) (x-3)/((x+1)(x-3))\\\\= \lim_(x \to 3)(1)/(x+1)\\ \\=(1)/(3+1)\\\\=(1)/(4)`

Hence, the limit of the expression:

`\lim_(x \to 3) (x-3)/(x^2-2x-3)` is:

`(1)/(4)`

Answer 2

Choice C is correct answer.

Explanation:

We have given expression.

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

We have to find the limit of function.

Simplifying the denominator, we have

x²-2x-3

Factoring the above expression, we have

(x-3)(x+1)

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

`\lim_(x \to \ 3) 1/x+1`

1 / 3+1

1/4

0.25

hence,
`\lim_(x \to \ 3) x-3/x^(2) -2x-3` = 0.25