Question

find $lim_{x
ightarrow c^{+}}f(x)$ for the given function and value of c.
$f(x)=left(\frac{7x}{x + 1}
ight)left(\frac{8x + 3}{x^{2}+x}
ight),c = 3$
$lim_{x
ightarrow3^{+}}left(\frac{7x}{x + 1}
ight)left(\frac{8x + 3}{x^{2}+x}
ight)=square$ (type an integer or a simplified fraction.)

Explanation:

Step1: Simplify the function

First, factor the denominator of the second - fraction: $x^{2}+x=x(x + 1)$. Then $f(x)=\frac{7x}{x + 1}\cdot\frac{8x + 3}{x(x + 1)}=\frac{7(8x + 3)}{(x + 1)^{2}}$.

Step2: Substitute $x = 3$

We want to find $\lim_{x
ightarrow3^{+}}\frac{7(8x + 3)}{(x + 1)^{2}}$. Substitute $x = 3$ into the function. When $x=3$, we have $\frac{7\times(8\times3 + 3)}{(3 + 1)^{2}}$.

Step3: Calculate the numerator and denominator

The numerator is $7\times(24 + 3)=7\times27 = 189$. The denominator is $(3 + 1)^{2}=16$.

Answer:

$\frac{189}{16}$