use the graph of the function ( f ) to decide whether each quantity exists. (if an answer does not exist…

use the graph of the function ( f ) to decide whether each quantity exists. (if an answer does not exist, enter dne.)\n(a) ( f(-2) )\n(b) ( lim_{x \to -2} f(x) )\n(c) ( f(0) )\n(d) ( lim_{x \to 0} f(x) )\n(e) ( f(2) )\n(f) ( lim_{x \to 2} f(x) )\n(g) ( f(4) )\n(h) ( lim_{x \to 4} f(x) )

use the graph of the function ( f ) to decide whether each quantity exists. (if an answer does not exist, enter dne.)\n(a) ( f(-2) )\n(b) ( lim_{x \to -2} f(x) )\n(c) ( f(0) )\n(d) ( lim_{x \to 0} f(x) )\n(e) ( f(2) )\n(f) ( lim_{x \to 2} f(x) )\n(g) ( f(4) )\n(h) ( lim_{x \to 4} f(x) )

Answer

Explanation:

Step1: Analyze ( f(-2) )

The function has a vertical asymptote at ( x = -2 ), so ( f(-2) ) does not exist.

Step2: Analyze ( \lim_{x\rightarrow - 2}f(x) )

As ( x ) approaches ( -2 ), the function values go to ( \pm\infty ), so ( \lim_{x\rightarrow - 2}f(x) ) does not exist (DNE).

Step3: Analyze ( f(0) )

From the graph, when ( x = 0 ), ( y=4 ), so ( f(0)=4 ).

Step4: Analyze ( \lim_{x\rightarrow0}f(x) )

As ( x ) approaches ( 0 ) from both sides, the function approaches ( y = 4 ), so ( \lim_{x\rightarrow0}f(x)=4 ).

Step5: Analyze ( f(2) )

The graph has an open - circle at ( x = 2 ), so ( f(2) ) does not exist (DNE).

Step6: Analyze ( \lim_{x\rightarrow2}f(x) )

As ( x ) approaches ( 2 ) from both sides, the function approaches ( y = 0 ), so ( \lim_{x\rightarrow2}f(x)=0 ).

Step7: Analyze ( f(4) )

From the graph, when ( x = 4 ), ( y=-1 ), so ( f(4)=-1 ).

Step8: Analyze ( \lim_{x\rightarrow4}f(x) )

As ( x ) approaches ( 4 ) from both sides, the function approaches ( y=\infty ), so ( \lim_{x\rightarrow4}f(x) ) does not exist (DNE).

Answer:

(a) DNE (b) DNE (c) (4) (d) (4) (e) DNE (f) (0) (g) (-1) (h) DNE