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