How do you find the limit #(1-t/(t-1))/(1-sqrt(t/(t-1))# as #x->oo#?

The given function is

To find the limit of the expression (1 - t/(t-1))/(1 - sqrt(t/(t-1))) as t approaches infinity, we can simplify the expression and then evaluate the limit.

First, let's simplify the expression: (1 - t/(t-1))/(1 - sqrt(t/(t-1)))

To simplify, we can multiply the numerator and denominator by the conjugate of the denominator, which is (1 + sqrt(t/(t-1))).

After simplifying, we get: [(1 - t/(t-1))(1 + sqrt(t/(t-1))))]/[(1 - sqrt(t/(t-1)))(1 + sqrt(t/(t-1)))]

Expanding and simplifying further, we have: [(1 - t/(t-1) + sqrt(t/(t-1)) - t/(t-1)*sqrt(t/(t-1)))]/[(1 - (t/(t-1)))/(1 - (t/(t-1)))]

Simplifying the denominator, we get: [(1 - t/(t-1) + sqrt(t/(t-1)) - t/(t-1)*sqrt(t/(t-1)))]/1

Further simplifying, we have: [1 - t/(t-1) + sqrt(t/(t-1)) - t/(t-1)*sqrt(t/(t-1))]

Now, let's evaluate the limit as t approaches infinity: As t approaches infinity, the terms involving t/(t-1) become negligible compared to the other terms. Therefore, we can ignore those terms.

Thus, the limit of the expression (1 - t/(t-1))/(1 - sqrt(t/(t-1))) as t approaches infinity is: 1 + sqrt(t/(t-1))

Therefore, the limit is 1.