Question

Lim x to infinity xe^(1/2x²) lhopistal rule:
A. 0
B. e¹/2
C. 1
D.[infinity]

Answer 1

The limit of
`xe^(1/2x²)` as x approaches infinity is found to be infinity after applying appropriate mathematical transformations to use L'Hôpital's rule.

Step-by-step explanation:

The question involves finding the limit of the function
`xe1/2x2` as x approaches infinity. To apply L'Hôpital's rule, we differentiate the numerator and the denominator with respect to x. However, in this case, we are not dealing with a fraction, so L'Hôpital's rule does not directly apply. The function is actually indeterminate of the form ∞ • 0, since
`e1/2x2`approaches 0 as x goes to infinity. Instead, we can use the substitution
`u = 1/2x2` which turns the function into
`(2u)eu`, and this is an indeterminate form 0 • ∞ where L'Hôpital's Rule can be applied after creating a fraction. Taking derivatives yields a limit of infinity, making the answer D. [infinity].