evaluate the limit\n\\(\\lim_{x\\to4}(x^{5}-7x + 7)\\)\nenter inf for \\(\\infty\\), -inf for…

Explanation:

Step1: Apply limit - sum rule

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

Step2: Apply constant - multiple rule and power rule

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

Step3: Calculate the result

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

Answer:

$1003$