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:
[tex]y-b=\dfrac{d-b}{c-a}\times (x-a)[/tex]
Here a,b)=(4,0) and (c,d)=(0,4)
Hence,
the equation of line is:
[tex]y-0=\dfrac{4-0}{0-4}\times (x-4)\\\\y=\dfrac{4}{-4}\tmes (x-4)\\\\y=-1(x-4)\\\\y=-x+4[/tex]
Now the left hand limit of the function at x=2 is:
[tex]\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[/tex]
Similarly the right hand limit of the function at x=2 is:
[tex]\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[/tex]
Hence, the limit of the function at x=2 is:
2
