Question

if 2x ≤ g(x) ≤ x^4 - x^2 + 2 for all x, evaluate lim(x→1) g(x). enter an exact number. resources read it

Explanation:

Step1: Find left - hand limit

Find $\lim_{x
ightarrow1}2x$. Using the direct - substitution property of limits, if $f(x)=2x$, then $\lim_{x
ightarrow1}2x = 2\times1=2$.

Step2: Find right - hand limit

Find $\lim_{x
ightarrow1}(x^{4}-x^{2}+2)$. Using the sum, difference, and power rules of limits:
$\lim_{x
ightarrow1}(x^{4}-x^{2}+2)=\lim_{x
ightarrow1}x^{4}-\lim_{x
ightarrow1}x^{2}+\lim_{x
ightarrow1}2$.
By the power rule $\lim_{x
ightarrow a}x^{n}=a^{n}$, we have $\lim_{x
ightarrow1}x^{4}=1^{4}=1$, $\lim_{x
ightarrow1}x^{2}=1^{2}=1$, and $\lim_{x
ightarrow1}2 = 2$.
So $\lim_{x
ightarrow1}(x^{4}-x^{2}+2)=1 - 1+2=2$.

Step3: Apply Squeeze Theorem

Since $2x\leq g(x)\leq x^{4}-x^{2}+2$ for all $x$ and $\lim_{x
ightarrow1}2x=\lim_{x
ightarrow1}(x^{4}-x^{2}+2) = 2$, by the Squeeze Theorem, $\lim_{x
ightarrow1}g(x)=2$.

Answer:

$2$