prove that cosine is a continuous function. we must show that $lim_{h\rightarrow0}cos(a + h)=cos(a)$ to…

prove that cosine is a continuous function. we must show that $lim_{h\rightarrow0}cos(a + h)=cos(a)$ to prove that the cosine function is continuous. $lim_{h\rightarrow0}cos(a + h)=lim_{h\rightarrow0}(cos(a)cdot\text{---select---} \text{?} sin(a)cdot\text{---select---})=(lim_{h\rightarrow0}cos(a))(lim_{h\rightarrow0}\text{---select---})\text{?} (lim_{h\rightarrow0}sin(a))(lim_{h\rightarrow0}\text{---select---})=(cos(a))(\text{?})\text{?} (sin(a))(\text{?})=cos(a)$ therefore, cosine is a continuous function.

prove that cosine is a continuous function. we must show that $lim_{h\rightarrow0}cos(a + h)=cos(a)$ to prove that the cosine function is continuous. $lim_{h\rightarrow0}cos(a + h)=lim_{h\rightarrow0}(cos(a)cdot\text{---select---} \text{?} sin(a)cdot\text{---select---})=(lim_{h\rightarrow0}cos(a))(lim_{h\rightarrow0}\text{---select---})\text{?} (lim_{h\rightarrow0}sin(a))(lim_{h\rightarrow0}\text{---select---})=(cos(a))(\text{?})\text{?} (sin(a))(\text{?})=cos(a)$ therefore, cosine is a continuous function.

Answer

Explanation:

Step1: Use cosine addition formula

$\lim_{h\rightarrow0}\cos(a + h)=\lim_{h\rightarrow0}(\cos(a)\cos(h)-\sin(a)\sin(h))$

Step2: Apply limit - product and difference rules

$=\left(\lim_{h\rightarrow0}\cos(a)\right)\left(\lim_{h\rightarrow0}\cos(h)\right)-\left(\lim_{h\rightarrow0}\sin(a)\right)\left(\lim_{h\rightarrow0}\sin(h)\right)$

Step3: Evaluate limits

Since $\cos(a)$ and $\sin(a)$ are constants with respect to $h$, $\lim_{h\rightarrow0}\cos(a)=\cos(a)$ and $\lim_{h\rightarrow0}\sin(a)=\sin(a)$. Also, $\lim_{h\rightarrow0}\cos(h) = 1$ and $\lim_{h\rightarrow0}\sin(h)=0$. So we have $(\cos(a))(1)-(\sin(a))(0)$

Step4: Simplify

$=\cos(a)$

Answer:

The cosine function is continuous as $\lim_{h\rightarrow0}\cos(a + h)=\cos(a)$ for all $a$.