Limits of Trigonometry

At the end of the lesson,
Limits of Exponential Functions
Limits of Trigonometry
Limits at Infinity
Infinite Limits
Evaluating the limits of the expressions 

 $\lim_{x\rightarrow0}\left(\frac{\sin\ x}{x}\right)=1$  
Example:
 $\lim_{x\rightarrow0}\left(\frac{\sin\ 3x}{x}\right)$  =  $\lim_{x\rightarrow0}\left(\frac{\sin\ 3x}{x}\right)\cdot\frac{3}{3}$  =  $\lim_{x\rightarrow0}\left(\frac{\sin\ 3x}{3x}\cdot3\right)$  =  $3$  

 $\lim_{x\rightarrow0}\left(\frac{1-\cos\ x}{x}\right)=0$  
Example:
 $\lim_{x\rightarrow0}\left(\frac{1-\cos\ 7x}{x}\right)$  =  $\lim_{x\rightarrow0}\left(\frac{1-\cos\ 7x}{x}\right)\cdot\frac{7}{7}$  =  $\lim_{x\rightarrow0}\left(\frac{1-\cos\ 7x}{7x}\cdot7\right)$  =  $0$  

 $\lim_{x\rightarrow0}\left(\frac{e^x-1}{x}\right)=1$  
Example:
 $\lim_{x\rightarrow0}\left(\frac{e^{2x}-1}{x}\right)$  =  $\lim_{x\rightarrow0}\left(\frac{e^{2x}-1}{x}\right)\cdot\frac{2}{2}$  =  $\lim_{x\rightarrow0}\ \ \left(\frac{e^{2x}-1}{2x}\right)\cdot2$  =  $2$