we are given that
f(x) is defined for all values of x except at x=c
Limit may or may not exist
case-1:
If there is hole at x=c , then limit exist
case-2:
If there is vertical asymptote at x=c , then limit does not exist
Examples:
case-1:
[tex] \lim_{x \to c} \frac{x^2-cx}{(x-c)}[/tex]
We can simplify it
[tex] \lim_{x \to c} \frac{x(x-c)}{(x-c)}[/tex]
[tex] =\lim_{x \to c} x[/tex]
[tex] =c[/tex]
so, we can see that limit exist and it's value defined
case-2:
[tex] \lim_{x \to c} \frac{1}{(x-c)}[/tex]
Left limit is
[tex] \lim_{x \to c-} \frac{1}{(x-c)}[/tex]
[tex] =-\infty[/tex]
Right Limit is
[tex] \lim_{x \to c+} \frac{1}{(x-c)}[/tex]
[tex] =+\infty[/tex]
so, we can see that left limit is not equal to right limit
so, limit does not exist
