Question

find lim f(x) for the given function and value of c. x→c⁺ f(x)=(7x / (x + 1))(8x + 3 / (x² + x)), c = 3 lim x→3⁺(7x / (x + 1))(8x + 3 / (x² + x))=□ (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: Calculate the right - hand limit

We want to find $\lim_{x
ightarrow3^{+}}\frac{7(8x + 3)}{(x + 1)^{2}}$. Substitute $x = 3$ into the simplified function. When $x=3$, we have $\frac{7\times(8\times3+3)}{(3 + 1)^{2}}$.
First, calculate the numerator: $8\times3+3=24 + 3=27$, and $7\times27 = 189$.
Then, calculate the denominator: $(3 + 1)^{2}=16$.

Answer:

$\frac{189}{16}$