# Basic Theroems on Limits

In this tutorial we give the statements of theorems on limits which will be useful in evaluating limits.

(1) The limit of a function, if it exists, is unique.

(2) If $m$ and $b$ are any constants, then $\mathop {\lim }\limits_{x \to a} \left( {mx + b} \right) = ma + b$.

(3) If $c$ is a constant, then for any number $a$, $\mathop {\lim }\limits_{x \to a} c = c$.

(4) For any constant $c$ and $\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$, we have

(5) The limit of the sum of two functions is equal to the sum of their limits, i.e. $\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$ and $\mathop {\lim }\limits_{x \to a} g\left( x \right) = M$, then

(6) The limit of the difference of two functions is equal to the difference of their limits, i.e. $\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$ and $\mathop {\lim }\limits_{x \to a} g\left( x \right) = M$, then

(7) The limit of the product of two functions is equal to the product of their limits, i.e. $\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$ and $\mathop {\lim }\limits_{x \to a} g\left( x \right) = M$, then

(8) The limit of the quotient of two functions is equal to the quotient of their limits, provided the limit of the denominator is not zero, i.e. $\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$ and $\mathop {\lim }\limits_{x \to a} g\left( x \right) = M$, then

(9) If $n$ is any integer and $\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$, then

Sandwich Theorem
If $g\left( x \right) \leqslant f\left( x \right) \leqslant h\left( x \right)$ for all numbers $x$ in an open interval containing a except possibly at a itself, such that

then