Question

Use the Squeeze Theorem to show that
`\lim_(t \to \infty) (cos5t)/(5t)`

Complete the following argument: for t arbitrary large and positive.

Im not sure how to break this problem down using the Squeeze Theorem, could anyone help?

Answer 1

Recall that
`\cos x` is bounded between -1 and 1 for all real
`x`, so that

`-1\le \cos(5t)\le 1`

`\implies-\frac1{5t}\le(\cos(5t))/(5t)\le\frac1{5t}`

Now take the limit of each side of the inequality:

`\displaystyle\lim_(t\to\infty)\left(-\frac1{5t}\right)\le\lim_(t\to\infty)(\cos(5t))/(5t)\le\lim_(t\to\infty)\frac1{5t}`

The bounding limits are both 0, so by the squeeze theorem the desired limit is also 0.