What is the limit of #6x^2 (cotx)(csc2x)# as #x->0#?

2 Answers
Aviv S.
Apr 2, 2018

The limit #lim_(xrarr0) 6x^2(cotx)(csc2x)=3#.

Explanation:

I'll assume you meant #lim_(xrarr0) 6x^2(cotx)(csc2x)#.

First, use trig identities to simplify down the limit:

#tanx=sinx/cosx#

#cotx=1/tanx=1/(sinx/cosx)=cosx/sinx#

#cscx=1/sinx#

#sin2x=2sinxcosx#

Now here's the problem:

#color(white)=lim_(xrarr0) 6x^2(cotx)(csc2x)#

#=lim_(xrarr0) 6x^2(cosx/sinx)(1/(sin2x))#

#=lim_(xrarr0) 6x^2(cosx/sinx)(1/(2sinxcosx))#

#=lim_(xrarr0) 6x^2(color(red)cancelcolor(black)cosx/sinx)(1/(2sinxcolor(red)cancelcolor(black)cosx))#

#=lim_(xrarr0) 6x^2(1/sinx)(1/(2sinx))#

#=lim_(xrarr0) 6x^2*1/2*(1/sin^2x)#

#=lim_(xrarr0) 3x^2*(1/sin^2x)#

#=lim_(xrarr0) (3x^2)/sin^2x#

Plugging in #x=0# yields #0/0#, so we can use L'Hospital's rule:

#=lim_(xrarr0) (3x^2)/sin^2x#

#=lim_(xrarr0) (d/dx(3x^2))/(d/dx(sin^2x))#

#=lim_(xrarr0) (2*3x)/(d/dx(sin^2x))#

#=lim_(xrarr0) (6x)/(d/dx(sinx*sinx))#

Product rule:

#=lim_(xrarr0) (6x)/(sinx*d/dx(sinx)+d/dx(sinx)*sinx)#

#=lim_(xrarr0) (6x)/(sinx*cosx+cosx*sinx)#

#=lim_(xrarr0) (6x)/(2sinxcosx)#

#=lim_(xrarr0) (6x)/(sin(2x))#

Plugging in #x=0# gives #0/0# again, so we can use L'Hospital's rule once more:

#=lim_(xrarr0) (6x)/(sin(2x))#

#=lim_(xrarr0) (d/dx(6x))/(d/dx(sin(2x)))#

#=lim_(xrarr0) (6)/(d/dx(sin(2x)))#

Chain rule:

#=lim_(xrarr0) (6)/(sin'(2x)*d/dx(2x))#

#=lim_(xrarr0) (6)/(cos(2x)*d/dx(2x))#

#=lim_(xrarr0) (6)/(cos(2x)*2)#

#=lim_(xrarr0) 3/cos(2x)#

#=3/cos(2*0)#

#=3/cos(0)#

#=3/1#

#=3#

That's the limit. We can verify this by looking at a graph:

graph{6x^2(cot(x))(csc(2x)) [-2.441, 2.425, 2.05, 4.483]}

Hope this helped!

George C.
Apr 2, 2018

#lim_(x->0) (6x^2 cot x csc 2x) = 3#

Explanation:

Here's a quick method using the Maclaurin series for #tan x# and #sin x#...

#lim_(x->0) (6x^2 cot x csc 2x) = lim_(x->0) (6x^2)/((tan x)(sin 2x))#

#color(white)(lim_(x->0) (6x^2 cot x csc 2x)) = lim_(x->0) (6x^2)/((x+O(x^3))(2x+O(x^3)))#

#color(white)(lim_(x->0) (6x^2 cot x csc 2x)) = lim_(x->0) 6/((1+O(x^2))(2+O(x^2)))#

#color(white)(lim_(x->0) (6x^2 cot x csc 2x)) = 3#

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