Question

Use the graph of the function f to determine the given limit. Picture below

Answer 1

Answer:

Option: a is correct.

Limit of the function at x=2 is: 2

Step-by-step explanation:

Clearly by looking at the graph of the function we could observe that the function f(x) is defined as:

f(x)=  -x+4   when x≠4

and    8 when x=2

since we could see that the function f(x) is a line segment that passes through the point (4,0) and (0,4).

and the equation of line passing through two points (a,b) and (c,d) is given by:

$y-b=\dfrac{d-b}{c-a}\times (x-a)$

Here a,b)=(4,0) and (c,d)=(0,4)

Hence,

the equation of line is:

$y-0=\dfrac{4-0}{0-4}\times (x-4)\\\\y=\dfrac{4}{-4}\tmes (x-4)\\\\y=-1(x-4)\\\\y=-x+4$

Now the left hand limit of the function at x=2 is:

$\lim_{h \to 0} f(2-h)\\\\= \lim_{h \to 0} -(2-h)+4\\ \\=\lim_{h \to 0} -2+h+4\\\\=\lim_{h \to 0}2+h\\\\=2$

Similarly the right hand limit of the function at x=2 is:

$\lim_{h \to 0} f(2+h)\\\\= \lim_{h \to 0} -(2+h)+4\\ \\=\lim_{h \to 0} -2-h+4\\\\=\lim_{h \to 0}2-h\\\\=2$

Hence, the limit of the function at x=2 is:

2