Find the limit?

Question

Find the limit of x^2*sin(1/x) as x approaches infinity.

 

Answer: ∞

 

Please show all your work. Thanks.

Answer 1

Find the limit of x^2*sin(1/x) as x approaches infinity.

 

-1 ≤ sin(1/x) ≤ 1 for all x

 

As x → +∞, sin(1/x) → sin(0⁺) → 0⁺

 

As x → –∞, sin(1/x) → sin(0^–) → 0^–

 

As x → ±∞, x^2 → +∞

 

So the sine function is going to contribute a sign to the result:

 

As x → +∞, x^2*sin(1/x) → +∞

 

As x → –∞, x^2*sin(1/x) → –∞

 

So the textbook was only considering x → +∞.

 

See http://www.wyzant.com/resources/files/264328/limit_x_squared_times_sine_of_one_over_x