You can put this solution on YOUR website! If you substitute a zero for x into your expression you end up with . This is an example of what is called an indeterminate form. A common way to find these limits is to use what is called L'Hopital's rule. This rule says for these indeterminate forms the limit of the ratios of the derivatives will be equal to the limit of original ratio.
So we find the derivatives of x and arctan(5x):
The derivative of x is 1. and the derivative of arctan(5x) (remembering to use the chain rule) is
We can now find the limit of the original ratio by finding the limit of the ratio of these derivatives:
lim
x->0
The fraction simplifies to:
lim
x->0
Substituting a zero for x we get a limit of 1/5.
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