Question

if (50x - 35x^{2}leq g(x)leq24 - 10x^{3}+x^{4}) for all (x), evaluate (lim_{x
ightarrow1}g(x)). limit

Explanation:

Step1: Find left - hand limit

Evaluate $\lim_{x
ightarrow1}(50x - 35x^{2})$.
Using limit rules $\lim_{x
ightarrow a}(f(x)+g(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)$ and $\lim_{x
ightarrow a}cx = c\lim_{x
ightarrow a}x$, $\lim_{x
ightarrow a}x^{n}=a^{n}$.
$\lim_{x
ightarrow1}(50x - 35x^{2})=50\lim_{x
ightarrow1}x-35\lim_{x
ightarrow1}x^{2}=50\times1 - 35\times1^{2}=50 - 35=15$.

Step2: Find right - hand limit

Evaluate $\lim_{x
ightarrow1}(24 - 10x^{3}+x^{4})$.
Using limit rules $\lim_{x
ightarrow a}(f(x)+g(x)+h(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)+\lim_{x
ightarrow a}h(x)$.
$\lim_{x
ightarrow1}(24 - 10x^{3}+x^{4})=24-10\lim_{x
ightarrow1}x^{3}+\lim_{x
ightarrow1}x^{4}=24-10\times1^{3}+1^{4}=24 - 10 + 1=15$.

Step3: Apply Squeeze Theorem

Since $\lim_{x
ightarrow1}(50x - 35x^{2})\leq\lim_{x
ightarrow1}g(x)\leq\lim_{x
ightarrow1}(24 - 10x^{3}+x^{4})$ and $\lim_{x
ightarrow1}(50x - 35x^{2})=\lim_{x
ightarrow1}(24 - 10x^{3}+x^{4}) = 15$, by the Squeeze Theorem, $\lim_{x
ightarrow1}g(x)=15$.

Answer:

$15$