Mathematica returns wrong limit

Question

I just started limits as part of my calculus class, and I have a simple limit to evaluate, $\mathop{\lim}\limits_{x \to 1}~f(x)$ . I found the limit to be 2 using the $f(x)=\mathop{\lim}\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$ method. However, upon checking my result in Mathematica (as well as Wolfram|Alpha), the result is shown as being 1. Is this a problem in Mathematica, or am I doing something wrong?

EDIT: $f(x)=x^2$

Answer 1

Well, for the function $f(x)=x^2$, we have both $$\lim_{x\to 1}\,f(x)=1\qquad\qquad \lim_{h\to 0}\,\frac{f(1+h)-f(1)}{h}=2.$$ Why would you expect them to give you the same answer?

Answer 2

f is defined at 1. Therefore $f(1)=1^2=1$

Now $f(x)\not =f'(x)=\mathop{\lim}\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$ and $f'(1)=2$