# 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
$\mathop {\lim }\limits_{x \to a} \left[ {cf\left( x \right)} \right] = c\mathop {\lim }\limits_{x \to a} f\left( x \right)$

(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
$\mathop {\lim }\limits_{x \to a} \left[ {f\left( x \right) + g\left( x \right)} \right] = \mathop {\lim }\limits_{x \to a} f\left( x \right) + \mathop {\lim }\limits_{x \to a} g\left( x \right) = L + M$

(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
$\mathop {\lim }\limits_{x \to a} \left[ {f\left( x \right) – g\left( x \right)} \right] = \mathop {\lim }\limits_{x \to a} f\left( x \right) – \mathop {\lim }\limits_{x \to a} g\left( x \right) = L – M$

(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
$\mathop {\lim }\limits_{x \to a} \left[ {f\left( x \right) \cdot g\left( x \right)} \right] = \mathop {\lim }\limits_{x \to a} f\left( x \right) \cdot \mathop {\lim }\limits_{x \to a} g\left( x \right) = L \cdot M$

(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
$\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}} = \frac{{\mathop {\lim }\limits_{x \to a} f\left( x \right)}}{{\mathop {\lim }\limits_{x \to a} g\left( x \right)}} = \frac{L}{M}$

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

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 $\mathop {\lim }\limits_{x \to a} g\left( x \right) = L = \mathop {\lim }\limits_{x \to a} h\left( x \right)$  then $\mathop {\lim }\limits_{x \to a} f\left( x \right) = L$