What is limit as theta ->0 of tan theta/theta

lim   tan(theta)/(theta)     as theta →  0  

lim  sin(theta) / [cos(theta)* (theta)]     as theta →  0  

lim [ sine(theta)/theta)] * [1/cos(theta)]  as theta →  0  

[1] * [1/1]   = 

1

Here's the graph showing this :  https://www.desmos.com/calculator/a6kkj0awy8

Find the following limit:
lim_(theta->0) (tan(theta))/theta

Using tan(theta) = (sin(theta))/(cos(theta)), write (tan(theta))/theta as (tan(theta))/theta:
lim_(theta->0) (sin(theta))/(theta cos(theta))

By the product rule,
lim_(theta->0) (tan(theta))/theta  =  (lim_(theta->0) (sin(theta))/theta) (lim_(theta->0) 1/(cos(theta))):
lim_(theta->0) 1/(cos(theta)) lim_(theta->0) (sin(theta))/theta

lim_(theta->0) sec(theta)  =  sec(0)  =  1:
lim_(theta->0) (sin(theta))/theta

Applying l'Hôpital's rule, we get that
lim_(theta->0) (sin(theta))/theta |  =  | lim_(theta->0) ( d/( dtheta) sin(theta))/(( dtheta)/( dtheta))
|  =  | lim_(theta->0) (cos(theta))/1
|  =  | lim_(theta->0) cos(theta)
lim_(theta->0) cos(theta)

lim_(theta->0) cos(theta)  =  cos(0)  =  1:
Answer: | = 1
 

Thanks so much! can you explain why 1/cos theta = 1?

Note ...... as theta →  0, cos(0) → 1  

.....so  lim  [ 1 / [cos(theta) ] as theta → 0   = 1/1   = 1

Here's a graph  of   [1/cos(theta)]  .......note the behavior  around 0  ....

https://www.desmos.com/calculator/xsprpg3jyi

Oooh okay, thank you guys so much! You're all super smart and lifesavers haha  <3