Question

Use the Squeeze Theorem to show that 5t%7D" id="TexFormula1" title="\lim_{t \to \infty} \frac{cos5t}{5t}" alt="\lim_{t \to \infty} \frac{cos5t}{5t}" align="absmiddle" class="latex-formula">
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

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

$\implies-\dfrac1{5t}\le\dfrac{\cos(5t)}{5t}\le\dfrac1{5t}$

Now take the limit of each side of the inequality:

$\displaystyle\lim_{t\to\infty}\left(-\dfrac1{5t}\right)\le\lim_{t\to\infty}\dfrac{\cos(5t)}{5t}\le\lim_{t\to\infty}\dfrac1{5t}$

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