Answer:
To make f continuous at [tex]x=2[/tex], [tex]f(2)[/tex] should be defined [tex]x = 2.[/tex]
Step-by-step explanation:
A function, let say [tex]f(x)[/tex], is defined at [tex]x = c[/tex] is continuous at [tex]x = c[/tex]
If the limit of [tex]f(x)[/tex] as [tex]x[/tex] approaches [tex]c[/tex] is equal to the value of [tex]f(x)[/tex] at [tex]x = c[/tex].
Mathematically it is written as:
if
[tex]\lim _{x\to c}\:f\left(x\right)=f\left(c\right)[/tex]
then
[tex]f(x)[/tex] is continuous at [tex]x = c[/tex].
So from the above definition, we conclude that:
To make f continuous at [tex]x=2[/tex], [tex]f(2)[/tex] should be defined [tex]x = 2.[/tex]
i.e.
if
[tex]\lim _{x\to 2}\:f\left(x\right)=f\left(2\right)[/tex]
then
[tex]f(x)[/tex] is continuous at [tex]x = 2[/tex].
To make [tex]f[/tex] continues at [tex]x=2[/tex] [tex]f(2)[/tex] should be defined at [tex]x=2[/tex]
Given-
Let [tex]f[/tex] be a real function subset of the real numbers and let [tex]c[/tex] be a point in the domain of function [tex]f[/tex]. Then [tex]f[/tex] is continuous at [tex]c[/tex] if,
[tex]\lim_{x-c} f(x)=f(c)[/tex]
when the limit of left hand, limit of right hand and the value of function at [tex]x=c[/tex] exist and are equal to each other, then function [tex]f[/tex] is said to be continues at [tex]x=c[/tex]. Its expression can be given as,
[tex]\lim_{x-c} f(x)=f(c)=\lim_{x-c'}[/tex]
Now for the given function [tex]f[/tex], to make [tex]f[/tex] continues at [tex]x=2[/tex]
[tex]\lim_{x-2} f(x)=f(2)[/tex]
thus function [tex]f(2)[/tex] should be defined at [tex]x=2[/tex].
Hence, To make [tex]f[/tex] continues at [tex]x=2[/tex] [tex]f(2)[/tex] should be defined at [tex]x=2[/tex]
For more about the continues function, follow the link below-
https://brainly.com/question/21447009
