find the limit of f(x) = \\frac{5}{x}-2 as x approaches \\infty and as x approaches -\\infty.\nlim f(x)…

Explanation:

Step1: Recall limit rule for $\frac{1}{x}$

As $x\rightarrow\pm\infty$, $\lim_{x\rightarrow\pm\infty}\frac{1}{x}=0$.

Step2: Rewrite the function for limit calculation

We have $f(x)=\frac{5}{x}-2 = 5\times\frac{1}{x}-2$.

Step3: Calculate the limit as $x\rightarrow\infty$

Using the limit - sum rule $\lim_{x\rightarrow a}(f(x)+g(x))=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$ and the constant - multiple rule $\lim_{x\rightarrow a}(cf(x)) = c\lim_{x\rightarrow a}f(x)$, we get $\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty}(\frac{5}{x}-2)=\lim_{x\rightarrow\infty}\frac{5}{x}-\lim_{x\rightarrow\infty}2$. Since $\lim_{x\rightarrow\infty}\frac{5}{x}=5\lim_{x\rightarrow\infty}\frac{1}{x}=0$ and $\lim_{x\rightarrow\infty}2 = 2$, then $\lim_{x\rightarrow\infty}f(x)=0 - 2=-2$.

Step4: Calculate the limit as $x\rightarrow-\infty$

Similarly, $\lim_{x\rightarrow-\infty}f(x)=\lim_{x\rightarrow-\infty}(\frac{5}{x}-2)=\lim_{x\rightarrow-\infty}\frac{5}{x}-\lim_{x\rightarrow-\infty}2$. Since $\lim_{x\rightarrow-\infty}\frac{5}{x}=5\lim_{x\rightarrow-\infty}\frac{1}{x}=0$ and $\lim_{x\rightarrow-\infty}2 = 2$, then $\lim_{x\rightarrow-\infty}f(x)=0 - 2=-2$.

Answer:

-2