Question

question 1: limit evaluation
evaluate the following limit. if the limit does not exist, enter dne. if the limit is infinite, specify if it is positive or negative infinity.
lim_{x
ightarrow0^{+}}left(\frac{5}{x^{2}+3x}-\frac{5}{3x}
ight).
lim_{x
ightarrow0^{+}}left(\frac{5}{x^{2}+3x}-\frac{5}{3x}
ight)=\text{symbolic expression}

Explanation:

Step1: Find a common - denominator

First, find a common denominator for the two fractions. The common denominator of $x^{2}+3x$ and $3x$ is $3x(x + 3)$. So, $\frac{5}{x^{2}+3x}-\frac{5}{3x}=\frac{5\times3-5(x + 3)}{3x(x + 3)}=\frac{15-5x - 15}{3x(x + 3)}=\frac{-5x}{3x(x + 3)}$.

Step2: Simplify the fraction

Cancel out the common factor $x$ in the numerator and denominator (since $x\to0^{+}$, $x
eq0$). The simplified fraction is $\frac{-5}{3(x + 3)}$.

Step3: Evaluate the limit

Now, find $\lim_{x\to0^{+}}\frac{-5}{3(x + 3)}$. Substitute $x = 0$ into the simplified function: $\lim_{x\to0^{+}}\frac{-5}{3(x + 3)}=\frac{-5}{3(0 + 3)}=-\frac{5}{9}$.

Answer:

$-\frac{5}{9}$