Question

Evaluate the limit using l'hospital's rule: lim x–> 0 (10ˣ-2ˣ)/x?

1) 0
2) 1
3) 2
4) Cannot be determined

Answer 1

The limit as x approaches 0 of (10^x - 2^x) / x using L'Hospital's Rule evaluates to the natural logarithm of 5, which is approximately 1.609 and not one of the provided answer choices.

Step-by-step explanation:

The limit in question is lim x→0 (10x - 2x) / x. We apply L'Hospital's Rule to this indeterminate form of type 0/0. To do so, we take the derivatives of the numerator and denominator separately with respect to x. The derivative of the numerator is ln(10)*10x - ln(2)*2x and the derivative of the denominator is simply 1. Upon substitution of x = 0 into the derivatives, we get ln(10)*1 - ln(2)*1, which simplifies to ln(5). Since this is a constant value, the limit evaluates to ln(5), which is approximately 1.609, and not any of the provided options (0, 1, 2, Cannot be determined).