Question

Using the definition of the limit at infinity verify that:

lim x-->infinity cos^2(x)/2x^2 = 0.

Answer 1

Conditions

Using the definition of the limit at infinity verify that $\lim x \to \infty$ if $x \to \infty$.

Solution

Consider the function $f(x)$ with a limit, equal to $F$, when $x \to \infty$:

$$\forall \varepsilon > 0 \, \exists \delta = \delta(\varepsilon) > 0 \, \forall x: |x| > \delta \, |f(x) - F| < \varepsilon$$

In our case $F = 0$. Let's verify, that the limit is equal to $F$.

For this let's fix $\varepsilon > 0$:

$$|f(x) - F| = \left| \frac{\cos^2(x)}{2x^2} - 0 \right| = \left| \frac{\cos^2(x)}{2x^2} \right| \leq \left| \frac{1}{2x^2} \right| < \left| \frac{1}{2\delta^2} \right| = \frac{1}{2\delta^2} < \varepsilon$$ $$\delta = \sqrt{\frac{1}{2\varepsilon}}$$

We've got, that

$$\forall \varepsilon > 0 \, \exists \delta = \delta(\varepsilon) = \sqrt{\frac{1}{2\varepsilon}} > 0 \, \forall x: |x| > \delta \, \left| \frac{\cos^2(x)}{2x^2} \right| < \varepsilon$$

Q.E.D.