Q. 13

Evaluate the following limits:


Answer :

As we need to find


We can directly find the limiting value of a function by putting the value of the variable at which the limiting value is asked if it does not take any indeterminate form (0/0 or ∞/∞ or ∞-∞, .. etc.)


Let Z


we need to take steps to remove this form so that we can get a finite value.


TIP: Most of the problems of logarithmic and exponential limits are solved using the formula and


This question is a direct application of limits formula of exponential and logarithmic limits and also use of sandwich theorem -


To get the desired forms, we need to include mx and nx as follows:


As Z


Z {Adding and subtracting 1 in numerator}


Z


{using algebra of limits}


To get the form as present in the formula we multiply and divide x into both terms respectively:


Z


{manipulating to get the forms present in formulae}


Z


Use the formula: and


Z


Hence,



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