Now, before we go to what Dubulus discussed, let us take a look on some properties of limits.

A. Suppose $\displaystyle \lim_{x\rightarrow a} f(x)$ exists. Then this limit is unique, i.e. if $\displaystyle \lim_{x\rightarrow a} f(x) = L_1$ and $\displaystyle \lim_{x\rightarrow a} f(x) = L_2$, then it must be $\displaystyle L_1 = L_2$.

B. The limit of a constant function is the constant itself, i.e. $\displaystyle \lim_{x\rightarrow a} c = c$.

C. Suppose $\displaystyle \lim_{x\rightarrow a} f(x) = L_1$ and $\displaystyle \lim_{x\rightarrow a} g(x) = L_2$. Then,

1. $\displaystyle \lim_{x\rightarrow a} \left(f(x)+g(x)\right) = L_1 + L_2$

2. $\displaystyle \lim_{x\rightarrow a} \left(f(x)-g(x)\right) = L_1 - L_2$

3. $\displaystyle \lim_{x\rightarrow a} \left(f(x)g(x)\right) = L_1 \cdot L_2$

4. $\displaystyle \lim_{x\rightarrow a} \left(\frac{f(x)}{g(x)}\right) = \frac{L_1}{L_2}$ if $\displaystyle g(a) \not=0$ and $\displaystyle L_2\not=0$.

other properties can be deduced from here.

Remark: If $\displaystyle f(x)$ is a function for which $\displaystyle f(a)$ exists, then $\displaystyle \lim_{x\rightarrow a} f(x) = f(a)$.

Theorems.

1. $\displaystyle \lim_{x\rightarrow a} f(x) = L \Longleftrightarrow \lim_{x\rightarrow a} f(x) - L = 0$

2. $\displaystyle \lim_{x\rightarrow a} f(x) = L \Longleftrightarrow \lim_{t\rightarrow 0} f(t + a) = L$

well, before i proceed, i have to know whether you still can catch up.