Special Limits Involving sin(x), x, and tan(x)

Key Questions

• How do you find the limit #lim_(x->0)sin(x)/x# ?

  We will use l'Hôpital's Rule.

  l'Hôpital's rule states:

  #lim_(x->a) f(x)/g(x) = lim_(x->a) (f'(x))/(g'(x))#

  In this example, #f(x)# would be #sinx#, and #g(x)# would be #x#.

  Thus,

  #lim_(x->0) (sinx)/x = lim_(x->0) (cosx)/(1)#

  Quite clearly, this limit evaluates to #1#, since #cos 0# is equal to #1#.

  Jacob F. · · Aug 18 2014
• How do you find the limit #lim_(x->0)tan(x)/x# ?

  Answer
  By using: #lim_{x to 0}{sinx}/{x}=1#, #lim_{x to 0}{tanx}/x=1#.

  Explanation
  Let us look at some details.
  Since #tanx={sinx}/{cosx}#,

  #lim_{x to 0}{tanx}/x = lim_{x to 0}{sinx}/{x}cdot1/{cosx}#

  by the Product Rule,

  #=(lim_{x to 0}{sinx}/x)cdot(lim_{x to 0}1/{cosx})#

  by #lim_{x to 0}{sinx}/{x}=1#,

  #=1cdot1/{cos(0)}=1#

  Wataru · · Sep 8 2014
• What are Special Limits Involving #y=sin(x)#?

  One of some useful limits involving #y=sin x# is
  #lim_{x to 0}{sin x}/{x}=1#.
  (Note: This limit indicates that two functions #y=sin x# and #y=x# are very similar when #x# is near #0#.)

  Wataru · · Sep 7 2014

• What are Special Limits Involving #y=sin(x)#?
• How do you find the limit #lim_(x->0)sin(x)/x# ?
• How do you find the limit #lim_(x->0)tan(x)/x# ?
• What is the derivative of #tanx^3#?
• What is the derivative of #tanx/x#?
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• How do you differentiate # g(x) =(1+cosx)/(1-cosx) #?
• What is the derivative of #tan(2x)#?
• How do you differentiate #f(x)=sinx/x#?
• How do you differentiate #f(x)=sinx/(1-cosx)#?
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• How do you differentiate #f(x)=tanx/(cosx-4)#?
• How do you differentiate #f(x)=cosx/(1+sinx)#?