Maths-
General
Easy
Question
- 1
- 0
- 2
- 4
Hint:
We can apply L'Hopital's rule, also commonly spelled L'Hospital's rule, whenever direct substitution of a limit yields an indeterminate form. This means that the limit of a quotient of functions (i.e., an algebraic fraction) is equal to the limit of their derivatives.
In this question, we have to find value of
.
The correct answer is: 2
![Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 2 x over denominator tan to the power of negative 1 end exponent space x end fraction](data:image/png;base64,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)
We first try substitution:
= ![Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 0 over denominator tan to the power of negative 1 end exponent space 0 end fraction space equals 0 over 0](data:image/png;base64,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)
Since the limit is in the form 0 over 0, it is indeterminate—we don’t yet know what is it. We need to do some work to put it in a form where we can determine the limit.
( We know that
)
On substituting, We get
= 2
Also, We can solve using L'Hospital's rule ;
.
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means
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