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