Basic Theroems on Limits

In this tutorial we give the statements of theorems on limits which will be useful in evaluating the limits.
(1) The limit of a function, if 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 nonzero, 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 some 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

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