Question

use the squeeze theorem to evaluate the limit $lim_{x
ightarrow8}f(x)$, if $16x - 64leq f(x)leq x^{2}$ on $6,10$. enter dne if the limit does not exist. limit =

Explanation:

Step1: Find the limit of the lower - bound function

We first find $\lim_{x
ightarrow8}(16x - 64)$. Using the limit rules for linear functions, $\lim_{x
ightarrow a}(mx + b)=ma + b$. Here $m = 16$ and $b=-64$, so $\lim_{x
ightarrow8}(16x - 64)=16\times8-64=128 - 64=64$.

Step2: Find the limit of the upper - bound function

Next, we find $\lim_{x
ightarrow8}x^{2}$. Using the power - rule for limits $\lim_{x
ightarrow a}x^{n}=a^{n}$, when $n = 2$ and $a = 8$, we have $\lim_{x
ightarrow8}x^{2}=8^{2}=64$.

Step3: Apply the Squeeze Theorem

Since $16x - 64\leq f(x)\leq x^{2}$ on the interval $[6,10]$ and $\lim_{x
ightarrow8}(16x - 64)=\lim_{x
ightarrow8}x^{2}=64$, by the Squeeze Theorem, $\lim_{x
ightarrow8}f(x)=64$.

Answer:

64