consider the following function.\n\n$f(x)=\\frac{5x}{x - 4}$\n\ndescribe the local behavior of the…

consider the following function.\n\n$f(x)=\\frac{5x}{x - 4}$\n\ndescribe the local behavior of the function.\n\n$f(x)$ is undefined at $x=$\n\n- as $x$ approaches this value from the left side,\n\n$f(x)$ approaches\n\n- as $x$ approaches this value from the right side,\n\n$f(x)$ approaches\n\ndescribe the end behavior of the function.\n\nas $x\\to\\infty$, $f(x)\\to$\n\nand as $x\\to-\\infty$, $f(x)\\to$\n\nquestion help: \$ \\square \$ message instructor\n\nsubmit question

Answer

Explanation:

Step1: Analyze the function ( f(x)=\frac{5x}{x - 4}) for local behavior at (x = 4)

Left - hand limit ((x\to4^{-})): Let (x=4 - h) where (h\to0^{+}). Then (f(x)=\frac{5(4 - h)}{(4 - h)-4}=\frac{20-5h}{-h}). As (h\to0^{+}), (f(x)\to-\infty).
Right - hand limit ((x\to4^{+})): Let (x = 4+h) where (h\to0^{+}). Then (f(x)=\frac{5(4 + h)}{(4 + h)-4}=\frac{20 + 5h}{h}). As (h\to0^{+}), (f(x)\to+\infty).

Step2: Analyze the end - behavior as (x\to\pm\infty)

Divide numerator and denominator by (x): (f(x)=\frac{5x}{x - 4}=\frac{5}{1-\frac{4}{x}}).
As (x\to\infty): (\lim_{x\to\infty}\frac{5}{1-\frac{4}{x}} = 5) (since (\lim_{x\to\infty}\frac{4}{x}=0)).
As (x\to-\infty): (\lim_{x\to-\infty}\frac{5}{1-\frac{4}{x}} = 5) (since (\lim_{x\to-\infty}\frac{4}{x}=0)).

Answer:

As (x) approaches (4) from the left side, (f(x)\to-\infty).
As (x) approaches (4) from the right side, (f(x)\to+\infty).
As (x\to\infty), (f(x)\to5).
As (x\to-\infty), (f(x)\to5).