Question

Given that lim x→a f(x) = 0 lim x→a g(x) = 0 lim x→a h(x) = 1 lim x→a p(x) = ∞ lim x→a q(x) = ∞, evaluate the limits below where possible. (if a limit is indeterminate, enter indeterminate.) (a) lim x→a [f(x) − p(x)] (b) lim x→a [p(x) − q(x)] (c) lim x→a [p(x) + q(x)]

Answer 1

Answer:

(a)  $L=-\infty$

(b)  $L=\text{Indeterminate}$

(c)   $L=\infty$

Step-by-step explanation:

We are given,

$\lim_{x\rightarrow a}f(x)=0$

$\lim_{x\rightarrow a}g(x)=0$

$\lim_{x\rightarrow a}h(x)=1$

$\lim_{x\rightarrow a}p(x)=\infty$

$\lim_{x\rightarrow a}q(x)=\infty$

Evaluate the limit:

(a) $L=\lim_{x\rightarrow a}[f(x)-p(x)]$

Using limit property, Distribute the limit

$L=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}p(x)]$

$L=0-\infty$

$L=-\infty$

(b) $L=\lim_{x\rightarrow a}[p(x)-q(x)]$

Using limit property, Distribute the limit

$L=\lim_{x\rightarrow a}p(x)-\lim_{x\rightarrow a}q(x)]$

$L=\infty-\infty$

$L=\text{Indeterminate}$

Because difference of two infinity might be finite or infinite.

(c) $L=\lim_{x\rightarrow a}[p(x)+q(x)]$

Using limit property, Distribute the limit

$L=\lim_{x\rightarrow a}p(x)+\lim_{x\rightarrow a}q(x)]$

$L=\infty+\infty$

$L=\infty$

Answer 2

Lim x→a f(x) = 0
lim x→a g(x) = 0
lim x→a h(x) = 1
lim x→a p(x) = ∞
lim x→a q(x) = ∞

(a) lim x→a [f(x) − p(x)]
lim x→a f(x) - lim x→a p(x)
0 - ∞
-∞

(b) lim x→a [p(x) − q(x)]
lim x→a p(x) - lim x→a q(x)
∞ - ∞
0

(c) lim x→a [p(x) + q(x)]
lim x→a p(x) + lim x→a q(x)
∞ + ∞
∞