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Question: If \[\displaystyle \lim_{x\rightarrow 3} \dfrac{x^{n}-3^{n}}{x-3} =108\], find the value of n....

If lim⁡x→3xn−3nx−3=108\displaystyle \lim_{x\rightarrow 3} \dfrac{x^{n}-3^{n}}{x-3} =108, find the value of n.

Explanation

Solution

Hint: In this question it is given that lim⁡x→3xn−3nx−3=108\displaystyle \lim_{x\rightarrow 3} \dfrac{x^{n}-3^{n}}{x-3} =108, we have to find the value of n. So to find the solution we need to know one formula, which is,
lim⁡x→axn−anx−a=nan−1\displaystyle \lim_{x\rightarrow a} \dfrac{x^{n}-a^{n}}{x-a} =na^{n-1}.....(1)
So by using the above formula we will solve the given equation.

Complete step-by-step solution:
Given,
lim⁡x→3xn−3nx−3=108\displaystyle \lim_{x\rightarrow 3} \dfrac{x^{n}-3^{n}}{x-3} =108
Now by the formula (1) we can write the above equation as,
⇒n⋅3n−1=108\Rightarrow n\cdot 3^{n-1}=108 [ where a=3]
⇒n⋅3n−1=4×27\Rightarrow n\cdot 3^{n-1}=4\times 27
⇒n⋅3n−1=4×33\Rightarrow n\cdot 3^{n-1}=4\times 3^{3}
⇒n⋅3n−1=4×34−1\Rightarrow n\cdot 3^{n-1}=4\times 3^{4-1}
So by comparing both sides we can easily say that, n=4.
Which is our required solution.
Note: While solving this type of question you need to know the basic formulas of limit that we have already mentioned in the hint portion, also we cannot solve the above equation by conventional method, so that is why we use the comparison method, i.e we have make same structure in both side of the equation and after that we can equate the corresponding values.