Warm up problem 20: Limits on trigo

Calculus Level 1

Evaluate

lim x 0 csc x tan x \displaystyle \lim_{x\rightarrow 0} \csc{x} ~ \tan{x}


The answer is 1.

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6 solutions

Trevor Arashiro
Nov 8, 2014

Well, you could use l'hopital's... Since csc (0) approaches \infty and tan (0) approaches 0,

But it's much simplier to use the identity csc x tan x= sec x. And sec(0)=1.

yes exactly

Mardokay Mosazghi - 6 years, 7 months ago
Parag Zode
Nov 9, 2014

We can solve it in this way :- l i m x 0 c o s e c x . t a n x lim_{x\rightarrow0}cosecx.tanx = l i m x 0 c o s e c x . s i n x c o s x lim_{x\rightarrow0}cosecx.\dfrac{sinx}{cosx} l i m x 0 1 s i n x s i n x c o s x lim_{x\rightarrow0}\dfrac{1}{sinx}\dfrac{sinx}{cosx} Cancelling sin x we get l i m x 0 1 c o s x lim_{x\rightarrow0}\dfrac{1}{cosx} l i m x 0 s e c x lim_{x\rightarrow0}secx now putting x = 0 x=0 we get, x = 1 x=1

Bhaskar Shriman
Nov 8, 2014

as x->0 sinx->x and tanx->x therefore csc x.tan x=tan x/sin x=x/x=1

Clarence Hife
Nov 11, 2014

cscxtanx=secx x=0; sec(0)= 1

Talha Butt
Nov 11, 2014

cosecx X tanx =1/ cosx ( cos0=1 ) so ans=1

Angellis Jose
Nov 10, 2014

since csc x tan x = 1/cos x and cos x = 1 as x approaches 0, so we'll get 1/1 =1

we may write cosec x inthe term of sin then expand sin and tan using x=0 as the pole using maclurins or tailorsexpansion and put x=0 to get ans 1 eaisly

nishant kumar Mishra - 6 years, 7 months ago

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