Calculation limit lim(x->1)tan(x-1)^2/x^2-1

To calculate the limit of the function as x approaches 1, we can use L'Hopital's Rule. This rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.

The function is:

f(x) = tan^2(x-1) / (x^2 - 1)

First, we need to find the derivatives of the numerator and the denominator.

The derivative of tan^2(x-1) is 2tan(x-1) * sec^2(x-1).

The derivative of x^2 - 1 is 2x.

Now, we can apply L'Hopital's Rule:

lim(x->1) [f'(x) / g'(x)] = lim(x->1) [2tan(x-1) * sec^2(x-1) / 2x]

Simplify the expression:

= lim(x->1) [tan(x-1) * sec^2(x-1) / x]

Now, substitute x = 1 into the expression:

= tan(1-1) * sec^2(1-1) / 1

= tan(0) * sec^2(0) / 1

= 0 * 1 / 1

= 0

So, the limit of the function as x approaches 1 is 0.