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

**Definition.** The function $f(x)$ has a limit, equal to $F$, when $x \to \infty$ in infinity, if:

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

Fix $\varepsilon > 0$.

**Consider** $|f(x) - F|$ for our case:

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

Here $\delta = \sqrt{\frac{1}{2\varepsilon}}$.

**So,** for each $\varepsilon$ we found $\delta = \delta(\varepsilon)$, for which $\left| \frac{\cos^2(x)}{2x^2} - 0 \right| < \varepsilon$.

Q.E.D.