How do you find the limit of #(cos (2x))^(3/(x^2))# as x approaches 0?

Answer 1

To find the limit of ( (\cos(2x))^{3/(x^2)} ) as ( x ) approaches 0, you can use the property of the natural logarithm to rewrite the expression as ( e^{\frac{3\ln(\cos(2x))}{x^2}} ). Then, apply L'Hôpital's Rule by taking the derivative of the numerator and the derivative of the denominator, and then finding the limit as ( x ) approaches 0. The result will be ( e^{\frac{-12}{0}} ), which tends towards ( e^{-\infty} ). Thus, the limit is 0.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Scarlett Allison

The limit should be #1/e^6#.
#lim_(x->0)cos^(3/x^2)(2x)=#

But:
#cos^(3/x^2)(2x)=e^[3/x^2ln[cos(2x)]# (have a look at the properties of logarithms)

and:
#lim_(x->0)e^[3/x^2ln[cos(2x)])=e^-6#

The exponent #3/x^2ln[cos(2x)]# tends to #-6#:

hope it is clear

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.