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  • If    <img alt="\lim _{x \rightarrow 2} \frac{x^n-2^n}{x-2}=80" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%202%7D%20%5Cfrac%7Bx%5En

If    \lim _{x \rightarrow 2} \frac{x^n-2^n}{x-2}=80    where n is a positive integer, then

Option: 1

0


Option: 2

2


Option: 3

5


Option: 4

None of these


Answers (1)

best_answer

Given,   

\lim _{x \rightarrow 2} \frac{x^n-2^n}{x-2}=80

By binomial theorem 

\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n \cdot a^{n-1}

Now L.H.S =\lim _{x \rightarrow 2} \frac{x^n-2^n}{x-2}=n .2^{n-1}

Then n .2^{n-1}=80

By taking n values  from options                  

a)     For n=3

      3.2^{3-1}=3.2^2=3(4)=12

b)     For n=2

      2.2^{2-1}=2.2^1=2(2)=4

c)     For n=5

      5.2^{5-1}=5.2^4=5(16)=80

So, for n=5

 {\lim _{x \rightarrow 2} \frac{x^n-2^n}{x-2}=80}

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