Question

1. calculate the following limits using continuity.

a. $lim_{\theta
ightarrowpi}\frac{cos(\theta)}{\theta^{4}+1}$
b. $lim_{t
ightarrow1}(arctan(t)cdot(t + 1)^{4})$

Explanation:

Step1: Recall continuity property

If $f(x)$ is continuous at $x = a$, then $\lim_{x
ightarrow a}f(x)=f(a)$.

Step2: Solve part A

The function $f(\theta)=\frac{\cos(\theta)}{\theta^{4}+1}$ is continuous for all real - valued $\theta$. So, $\lim_{\theta
ightarrow\pi}\frac{\cos(\theta)}{\theta^{4}+1}=\frac{\cos(\pi)}{\pi^{4}+1}$. Since $\cos(\pi)= - 1$, we have $\frac{-1}{\pi^{4}+1}=-\frac{1}{\pi^{4}+1}$.

Step3: Solve part B

The function $g(t)=\arctan(t)\cdot(t + 1)^{4}$ is continuous at $t = 1$. So, $\lim_{t
ightarrow1}(\arctan(t)\cdot(t + 1)^{4})=\arctan(1)\cdot(1 + 1)^{4}$. Since $\arctan(1)=\frac{\pi}{4}$ and $(1 + 1)^{4}=16$, we get $\frac{\pi}{4}\times16 = 4\pi$.

Answer:

A. $-\frac{1}{\pi^{4}+1}$
B. $4\pi$