Question

evaluate the limit
\\(\lim_{x\to4}(x^{5}-7x + 7)\\)
enter inf for \\(\infty\\), -inf for \\(-\infty\\), and dne if the limit does not exist.
limit =

Explanation:

Step1: Apply limit - sum rule

$\lim_{x
ightarrow a}(f(x)+g(x)+h(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)+\lim_{x
ightarrow a}h(x)$
So, $\lim_{x
ightarrow 4}(x^{5}-7x + 7)=\lim_{x
ightarrow 4}x^{5}-\lim_{x
ightarrow 4}7x+\lim_{x
ightarrow 4}7$

Step2: Apply constant - multiple rule and power rule

For $\lim_{x
ightarrow 4}7x$, by constant - multiple rule $\lim_{x
ightarrow a}cf(x)=c\lim_{x
ightarrow a}f(x)$, we have $\lim_{x
ightarrow 4}7x = 7\lim_{x
ightarrow 4}x$. And by power rule $\lim_{x
ightarrow a}x^{n}=a^{n}$, $\lim_{x
ightarrow 4}x^{5}=4^{5}$ and $\lim_{x
ightarrow 4}x = 4$. Also, for a constant function $y = c$, $\lim_{x
ightarrow a}c=c$, so $\lim_{x
ightarrow 4}7 = 7$.
$7\lim_{x
ightarrow 4}x=7\times4$, $\lim_{x
ightarrow 4}x^{5}=4^{5}=1024$

Step3: Calculate the result

$\lim_{x
ightarrow 4}(x^{5}-7x + 7)=4^{5}-7\times4 + 7=1024-28 + 7=996 + 7=1003$

Answer:

$1003$