Question

Find the value of √6/√x when xⁿ-aⁿ=0​

Answer 1

Step-by-step explanation:

So basically you want to solve the limit (x tends to a) of (xn−an)/(x−a) .

Clearly we can see that initially it is of the form 0/0 when we plug in x=a .

Now we have two methods to do it.

One way is to divide (xn−an) by (x−a) and then when you’ve got rid of the denominator, plug in x=a and get the final answer. This can be done by mathematical induction as well as by binomial expansion.

So xn−an can be written as (x−a)(xn−1+a.xn−2+a2.xn−3+…+an−2.x+an−1)

On dividing (xn−an) by (x−a) we cancel (x−a) from the above expansion and the rest of the expression can then be evaluated by plugging in x=a as it is no longer 0/0 form.

So there are n terms in total and each term will be an−1 , giving us the result n.an−1 .

The other method to do it is to recall the L’Hopital’s Rule. Whenever the limit evaluates directly to 0/0 form or infinity/infinity form, we differentiate the numerator and denominator separately wrt the dependent variable until the limit is no longer 0/0 or infinity/infinity form.

So differentiating the numerator wrt x gives us n.xn−1 and differentiating the denominator wrt x gives us 1 . Now we plug in x=a and get the same answer n.an−1 .

I hope that is what you were looking for.