consider the following function. f(x) = \\frac{5x}{x - 7} describe the local behavior of the function. f(x)…

consider the following function. f(x) = \\frac{5x}{x - 7} describe the local behavior of the function. f(x) is undefined at x = 7. as x approaches this value from the left side, f(x) approaches. as x approaches this value from the right side, f(x) approaches. describe the end behavior of the function. as x → ∞, f(x) → and as x → -∞, f(x) →. question help: message instructor

consider the following function. f(x) = \\frac{5x}{x - 7} describe the local behavior of the function. f(x) is undefined at x = 7. as x approaches this value from the left side, f(x) approaches. as x approaches this value from the right side, f(x) approaches. describe the end behavior of the function. as x → ∞, f(x) → and as x → -∞, f(x) →. question help: message instructor

Answer

Explanation:

Step1: Analyze left - hand limit

We want to find $\lim_{x\rightarrow7^{-}}\frac{5x}{x - 7}$. Let $x=7 - h$, where $h>0$ and $h\rightarrow0$. Then $f(7 - h)=\frac{5(7 - h)}{(7 - h)-7}=\frac{35-5h}{-h}$. As $h\rightarrow0$, $\frac{35-5h}{-h}\rightarrow-\infty$.

Step2: Analyze right - hand limit

We want to find $\lim_{x\rightarrow7^{+}}\frac{5x}{x - 7}$. Let $x = 7+h$, where $h>0$ and $h\rightarrow0$. Then $f(7 + h)=\frac{5(7 + h)}{(7 + h)-7}=\frac{35 + 5h}{h}$. As $h\rightarrow0$, $\frac{35+5h}{h}\rightarrow\infty$.

Step3: Analyze end - behavior

For $\lim_{x\rightarrow\pm\infty}\frac{5x}{x - 7}$, divide both numerator and denominator by $x$: $\lim_{x\rightarrow\pm\infty}\frac{5x/x}{(x - 7)/x}=\lim_{x\rightarrow\pm\infty}\frac{5}{1-\frac{7}{x}}$. As $x\rightarrow\pm\infty$, $\frac{7}{x}\rightarrow0$, so $\lim_{x\rightarrow\pm\infty}\frac{5}{1-\frac{7}{x}} = 5$.

Answer:

As $x$ approaches $7$ from the left side, $f(x)$ approaches $-\infty$. As $x$ approaches $7$ from the right side, $f(x)$ approaches $\infty$. As $x\rightarrow\infty$, $f(x)\rightarrow5$. As $x\rightarrow-\infty$, $f(x)\rightarrow5$.