Limits of Trigonometry

Limits (Part 2)

Basic Calculus

Objectives

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

 $\frac{\sin x}{x}$ ,   $\frac{1-\cos x}{x}$  ,  $\frac{e^x-1}{x}$  

Multiple Choice

Find  $\lim_{x\rightarrow0}\sin\ x\ +\ 1$  

• 0

• 1

• -1

• 2

Multiple Choice

Find  $\lim_{x\rightarrow0}\left(\sin\left(\frac{x^2-1}{x-1}\right)\right)$  

• 2

• 0.017

• 1

• 0.035

Multiple Choice

Find  $\lim_{x\rightarrow0}\tan\ x+\sin2x$  

• 1

• 2

• 0

• 0.5

Multiple Choice

 $\lim_{x\rightarrow0}\left(\frac{1}{x}\right)$  

• 0

• undefined

• Does not exist

•  $\infty$  
  

Multiple Choice

 $\lim_{x\rightarrow-1}\frac{x+3}{\left(x+1\right)^2}$  

• undefined

• 0

• DNE

•  $\infty$  

Multiple Choice

 $\lim_{x\rightarrow-6}\left(\frac{2x}{6+x}\right)$  

• Does not exist

• 0

•  $\infty$  

• undefined

Multiple Choice

 $\lim_{x\rightarrow2}\left(\frac{x+7}{x^2-4}\right)$  

•  $\infty$  

•  $0$  

• Does not exist

• undefined

Multiple Choice

 $\lim_{x\rightarrow0}\frac{7x}{\left(10-3x\right)^4}$  

•  $0$  

•  $\infty$  

• Does not exist

• undefined

Multiple Choice

 $\lim_{x\rightarrow3}\frac{9}{\left(x-3\right)^4}$  

• Does not exist

•  $0$  

•  $\infty$  

• undefined
  
  

Multiple Choice

 $\lim_{x\rightarrow-5\ \ \ }-\frac{8}{\left(x+5\right)^2}$  

• undefined

•  $\infty$  

• Does not exist

•  $-\infty$  

Multiple Choice

 $\lim_{x\rightarrow\infty}4x^7-18x^3+9$  

•  $9$ 

• Does not exist
  

•  $-\infty$  

•  $\infty$  

Multiple Choice

 $\lim_{x\rightarrow-\infty}3xe^x$  

• undefined

•  $-\infty$  

• Does not exist

•  $\infty$  

Special Limits

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

Special Limits

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

Special Limits

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