find $lim_{x\rightarrow c^{+}}f(x)$ for the given function and value of c.\n$f(x)=left(\frac{4x}{x +…

find $lim_{x\rightarrow c^{+}}f(x)$ for the given function and value of c.\n$f(x)=left(\frac{4x}{x + 1}\right)left(\frac{7x + 2}{x^{2}+x}\right),c = 2$\n$lim_{x\rightarrow2^{+}}left(\frac{4x}{x + 1}\right)left(\frac{7x + 2}{x^{2}+x}\right)=square$\n(type an integer or a simplified fraction.)

find $lim_{x\rightarrow c^{+}}f(x)$ for the given function and value of c.\n$f(x)=left(\frac{4x}{x + 1}\right)left(\frac{7x + 2}{x^{2}+x}\right),c = 2$\n$lim_{x\rightarrow2^{+}}left(\frac{4x}{x + 1}\right)left(\frac{7x + 2}{x^{2}+x}\right)=square$\n(type an integer or a simplified fraction.)

Answer

Explanation:

Step1: Simplify the function

First, factor the denominator of the second - fraction: $x^{2}+x=x(x + 1)$. Then $f(x)=\left(\frac{4x}{x + 1}\right)\left(\frac{7x+2}{x(x + 1)}\right)=\frac{4(7x + 2)}{(x + 1)^{2}}$.

Step2: Calculate the right - hand limit

We want to find $\lim_{x\rightarrow2^{+}}\frac{4(7x + 2)}{(x + 1)^{2}}$. Substitute $x = 2$ into the simplified function. When $x=2$, we have $\frac{4\times(7\times2+2)}{(2 + 1)^{2}}$. First, calculate the numerator: $4\times(7\times2+2)=4\times(14 + 2)=4\times16 = 64$. Then, calculate the denominator: $(2 + 1)^{2}=9$.

Answer:

$\frac{64}{9}$