Estimate the limit. Picture below

Question

asked May 18, 2020 211k views

2 Answers

Jozcar · Answer 1 · 2020-05-23T18:07:07+0000

Answer:

Hence, the limit of the expression:
$\lim_(x \to 1) ((1)/(x+2)-(1)/(3))/(x-1)$ is:

(-1)/(9)

Explanation:

We are asked to estimate the limit of the expression:

$\lim_(x \to 1) ((1)/(x+2)-(1)/(3))/(x-1)$

We will simplify the expression by first taking the l.c.m of the terms in the numerator to obtain the expression as:

$(3-(x+2))/(3(x+2))\\\\=(3-x-2)/(3(x+2))\\\\=(1-x)/(3(x+2))$

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

since the same term in the numerator and denominator are cancelled out.

Now the limit of the function exist as the denominator is not equal to zero when x→1.

Hence,

$\lim_(x \to 1) (-1)/(3(x+2))\\\\=(-1)/(3(1+2))\\\\=(-1)/(3* 3)\\\\=(-1)/(9)$

Hence, the limit of the expression:
$\lim_(x \to 1) ((1)/(x+2)-(1)/(3))/(x-1)$ is: