Question

given that $lim_{x
ightarrow2}f(x)= - 4$ and $lim_{x
ightarrow2}g(x)=4$, find the following limit.
$lim_{x
ightarrow2}\frac{5 - f(x)}{x + g(x)}$
$lim_{x
ightarrow2}\frac{5 - f(x)}{x + g(x)}=square$ (simplify your answer.)

Explanation:

Step1: Apply limit - sum and difference rules

By the rules of limits, $\lim_{x
ightarrow a}(u(x)\pm v(x))=\lim_{x
ightarrow a}u(x)\pm\lim_{x
ightarrow a}v(x)$ and $\lim_{x
ightarrow a}\frac{u(x)}{v(x)}=\frac{\lim_{x
ightarrow a}u(x)}{\lim_{x
ightarrow a}v(x)}$ (where $\lim_{x
ightarrow a}v(x)
eq0$). So, $\lim_{x
ightarrow2}\frac{5 - f(x)}{x + g(x)}=\frac{\lim_{x
ightarrow2}(5 - f(x))}{\lim_{x
ightarrow2}(x + g(x))}$.

Step2: Apply limit - sum and difference rules again

$\lim_{x
ightarrow2}(5 - f(x))=\lim_{x
ightarrow2}5-\lim_{x
ightarrow2}f(x)$ and $\lim_{x
ightarrow2}(x + g(x))=\lim_{x
ightarrow2}x+\lim_{x
ightarrow2}g(x)$. We know that $\lim_{x
ightarrow2}5 = 5$, $\lim_{x
ightarrow2}f(x)=-4$, $\lim_{x
ightarrow2}x = 2$, and $\lim_{x
ightarrow2}g(x)=4$.

Step3: Calculate the numerator and denominator

The numerator $\lim_{x
ightarrow2}5-\lim_{x
ightarrow2}f(x)=5-(-4)=5 + 4=9$. The denominator $\lim_{x
ightarrow2}x+\lim_{x
ightarrow2}g(x)=2 + 4=6$.

Step4: Find the limit value

$\lim_{x
ightarrow2}\frac{5 - f(x)}{x + g(x)}=\frac{9}{6}=\frac{3}{2}$.

Answer:

$\frac{3}{2}$