Consider the given limits (a is a constant). lim x ? a f ( x ) = 0 lim x ? a g ( x ) = 0 lim...

Question:

Consider the given limits (a is a constant).

{eq}\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 {/eq}

Evaluate each limit below. If a limit is indeterminate, enter INDETERMINATE.

(a) {eq}\lim_{x\rightarrow a} [f(x)]^{g(x)} {/eq}

(b) {eq}\lim_{x\rightarrow a} [f(x)]^{p(x)} {/eq}

(c) {eq}\lim_{x\rightarrow a} [h(x)]^{p(x)} {/eq}

(d) {eq}\lim_{x\rightarrow a} [p(x)]^{f(x)} {/eq}

Limits:

Limits, when applied, determine the values of expressions of functions at a specific location, denoted as {eq}a {/eq}. The limit shows that as we get closer and closer to {eq}a {/eq}, the expression of function approaches a specific value.

Given: {eq}\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 \\ {/eq}

a.

{eq}\begin{align*} \lim_{x\rightarrow a} [f(x)]^{g(x)} &= \lim_{x\rightarrow a} [0]^{0} \\ &= \text{INDETERMINATE} \\ \end{align*} \\ {/eq}

b.

{eq}\begin{align*} \lim_{x\rightarrow a} [f(x)]^{p(x)} &= \lim_{x\rightarrow a} [0]^{\infty} \\ &= 0 \\ \end{align*}\\ {/eq}

c.

{eq}\begin{align*} \lim_{x\rightarrow a} [h(x)]^{p(x)} &= \lim_{x\rightarrow a} [1]^{\infty} \\ &= \text{INDETERMINATE} \\ \end{align*}\\ {/eq}

d.

{eq}\begin{align*} \lim_{x\rightarrow a} [p(x)]^{f(x)} &= \lim_{x\rightarrow a} [\infty]^{0} \\ &= \text{INDETERMINATE} \\ \end{align*} {/eq}