Question

find the limit, if it exists. (if an answer does not exist, enter dne.)

$$\\lim_{x\\to -\\infty}\\left(\\frac{9}{x}-\\frac{x}{5}\ ight)$$

Explanation:

Step1: Split the limit

By the property of limits \(\lim_{x
ightarrow a}(f(x)-g(x))=\lim_{x
ightarrow a}f(x)-\lim_{x
ightarrow a}g(x)\), we have \(\lim_{x
ightarrow-\infty}(\frac{9}{x}-\frac{x}{5})=\lim_{x
ightarrow-\infty}\frac{9}{x}-\lim_{x
ightarrow-\infty}\frac{x}{5}\).

Step2: Evaluate \(\lim_{x

ightarrow-\infty}\frac{9}{x}\)
As \(x
ightarrow-\infty\), for the function \(y = \frac{9}{x}\), using the formula \(\lim_{x
ightarrow\pm\infty}\frac{c}{x}=0\) (\(c\) is a constant), we get \(\lim_{x
ightarrow-\infty}\frac{9}{x}=0\).

Step3: Evaluate \(\lim_{x

ightarrow-\infty}\frac{x}{5}\)
As \(x
ightarrow-\infty\), for the function \(y=\frac{x}{5}\), since the coefficient of \(x\) is \(\frac{1}{5}\), we have \(\lim_{x
ightarrow-\infty}\frac{x}{5}=-\infty\).

Step4: Calculate the final limit

Substitute the results of Step 2 and Step 3 into the expression from Step 1: \(0-(-\infty)=\infty\).

Answer:

\(\infty\)