If lim _(xarrow a)(a^3-5a^2x)/(a^2)-25x^(2)=lim _(xarrow 1)(x^4-1)/(x^3)-1 then find a. Rationalization

Question

Answer 1

To solve this problem, we need to find the value of

such that the two limits are equal.Given:

First, let's evaluate the right-hand side limit:

We can factorize the numerator and denominator:

Thus,

Cancel the common factor \((x - 1)\):

Now, evaluate the limit as

:

So, we have:

Next, let's evaluate the left-hand side limit:

Factorize the numerator and denominator:

Thus,

Now, evaluate the limit as

:

Set this equal to

:

Solve for

:

Square both sides:

This is a quadratic equation in

. Solving it will give us the value of

. After solving, we find:

Thus, the value of

is

.

Question