- #1

nycmathdad

- 74

- 0

lim [x^2 • (1 - cos(1/x)]

x--> 0

Let me see.

-1 ≤ cos (1/x) ≤ 1

-x^2 ≤ x^2 • [1 - cos(1/x)] ≤ x^2

-|x^2| ≤ x^2 • [1 - cos(1/x)] ≤ |x^2|

lim -|x^2| as x tends to 0 = 0.

lim |x^2| as x tends to 0 = 0.

.

By the Squeeze Theorem, [x^2 • (1 - cos(1/x)] was squeezed between the limit of -|x^2| as x tends to 0 and the limit of |x^2| as x tends to 0.

Conclusion:

lim [x^2 •(1 - cos(1/x)] = 0

x--> 0

The limit is 0.

Correct?