Question

Find the limit of the function lim t→0 4et - 4 t , 1 t - 1 t , 3 1 t?

Answer 1

To calculate the limit, we can use L'Hôpital's Rule. Taking the derivative of the numerator and denominator separately, we simplify the expression and substitute t = 0 to find that the limit is 0.

Step-by-step explanation:

The given limit is:

limt→0 4et - 4t / (1 - t)-1 / 3 - 1 / t

To find this limit, we can use L'Hôpital's Rule. Taking the derivative of the numerator and denominator separately:

1. Derivative of 4et - 4t w.r.t. t: 4et - 4
2. Derivative of (1 - t)-1 / 3 - 1 / t w.r.t. t: [(0 - 1)(1 - t)-1 - 1(0 - t)-1] / [(1 - t)-2 - t-2]

Substituting t = 0 into both derivatives, the limit simplifies to:

limt→0 4et - 4 / 1 / (3 - 1)

Further simplifying the expression, we get:

limt→0 4(et - 1) / 2

Now substituting t = 0:

4(e0 - 1) / 2 = 4(1 - 1) / 2 = 0