. . Indeterminate forms ∞ ∞ ×∞∞−∞ Example . Evaluate : lim .
If we put directly x = we observe that the given function is in an indeterminate form . As the numerator and the denominator functions are polynomials of degree they both are differentiable. Hence, by an application of the L’Hôpital’s Rule, we get lim = lim = . Note that this limit may also be evaluated through the factorization of the numerator and denominator as x )( )( ) .
Example . Compute the limit lim . If we put directly x we observe that the given function is in an indeterminate form . As the numerator and the denominator functions are polynomials they both are differentiable.
Applications of Differential Calculus Hence by an application of the L’Hôpital’s Rule we get, lim = lim × = n a n × − . Example . Evaluate the limit lim sin mx If we directly substitute x = we get an indeterminate form and hence we apply the L’Hôpital’s rule to evaluate the limit as, lim sin mx = lim mx × = m The next example tells that the limit does not exist. Example .
Evaluate the limit lim sin If we directly substitute x = we get an indeterminate form and hence we apply the L’Hôpital’s rule to evaluate the limit as, lim sin → + = lim → + = ∞ lim sin → − = lim → − = −∞ As the left limit and the right limit are not the same we conclude that the limit does not exist. Remark One may be tempted to use the L’Hôpital’s rule once again in lim → + to conclude lim → + = lim → + = . which is not true because it was not an indeterminate form. Example .