Question

given $lim_{x
ightarrow4}f(x)=8$ and $lim_{x
ightarrow4}g(x)=7$, evaluate $lim_{x
ightarrow4}\frac{f(x)+g(x)}{7f(x)}$. (if the limit does not exist, enter \dne\). limit =

Explanation:

Step1: Apply limit - sum and quotient rules

By the sum rule of limits $\lim_{x
ightarrow a}(f(x)+g(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)$ and the quotient rule $\lim_{x
ightarrow a}\frac{u(x)}{v(x)}=\frac{\lim_{x
ightarrow a}u(x)}{\lim_{x
ightarrow a}v(x)}$ (where $\lim_{x
ightarrow a}v(x)
eq0$), we have $\lim_{x
ightarrow 4}\frac{f(x)+g(x)}{7f(x)}=\frac{\lim_{x
ightarrow 4}(f(x)+g(x))}{\lim_{x
ightarrow 4}(7f(x))}$.

Step2: Apply constant - multiple rule

The constant - multiple rule of limits states that $\lim_{x
ightarrow a}(cf(x)) = c\lim_{x
ightarrow a}f(x)$ for a constant $c$. So, $\lim_{x
ightarrow 4}(7f(x))=7\lim_{x
ightarrow 4}f(x)$ and $\lim_{x
ightarrow 4}(f(x)+g(x))=\lim_{x
ightarrow 4}f(x)+\lim_{x
ightarrow 4}g(x)$.

Step3: Substitute the given limit values

We know that $\lim_{x
ightarrow 4}f(x) = 8$ and $\lim_{x
ightarrow 4}g(x)=7$. Substituting these values, we get $\frac{\lim_{x
ightarrow 4}f(x)+\lim_{x
ightarrow 4}g(x)}{7\lim_{x
ightarrow 4}f(x)}=\frac{8 + 7}{7\times8}$.

Step4: Simplify the expression

$\frac{8 + 7}{7\times8}=\frac{15}{56}$.

Answer:

$\frac{15}{56}$