exercises 2.3 the limit laws\nscore: 18.75/21 answered: 19/21\nquestion 20\ntextbook videos +\nfind the…

Explanation:

Step1: Recall tangent identity

We know that $\tan(7x)=\frac{\sin(7x)}{\cos(7x)}$, so the limit becomes $\lim_{x\rightarrow0}\frac{\sin(7x)}{x\cos(7x)}$.

Step2: Rewrite the limit

$\lim_{x\rightarrow0}\frac{\sin(7x)}{x\cos(7x)}=\lim_{x\rightarrow0}\frac{\sin(7x)}{x}\cdot\lim_{x\rightarrow0}\frac{1}{\cos(7x)}$.

Step3: Use the limit - rule $\lim_{u\rightarrow0}\frac{\sin u}{u} = 1$

Let $u = 7x$. As $x\rightarrow0$, $u\rightarrow0$. And $\lim_{x\rightarrow0}\frac{\sin(7x)}{x}=7\lim_{x\rightarrow0}\frac{\sin(7x)}{7x}=7\times1 = 7$.

Step4: Evaluate $\lim_{x\rightarrow0}\frac{1}{\cos(7x)}$

Since $\cos(0)=1$, $\lim_{x\rightarrow0}\frac{1}{\cos(7x)}=\frac{1}{\cos(0)} = 1$.

Step5: Calculate the original limit

$\lim_{x\rightarrow0}\frac{\sin(7x)}{x}\cdot\lim_{x\rightarrow0}\frac{1}{\cos(7x)}=7\times1=7$.

Answer:

$7$