1) Suppose that a function f(x) is defined for all real values of x, except x = xo. Can anything be said about LaTeX: \displaystyle\lim\limits_{x\to x_0} f(x)lim x → x 0 f ( x )? Give reasons for your answer.

Ok, we have that f(x) is defined for all real values of x, except for x = x0.

Does it exist? why?

Remember that when we are taking the limit we are not evaluating the function in x0, instead, we are evaluating the function in values really close to x0 (values defined as x0⁺ and x0⁻, where the sign defines if we approach from above or bellow).

And because f(x) is defined in the values of x near x0, we can conclude that  the limit does exist if:

if that does not happen, like in f(x) = 1/x where x0 = 0

where the lower limit is negative and the upper limit is positive, we have that the limit does not converge.