How do you find f'(2) using the limit definition given #3/(x+1)#?

1 Answer
mason m
Aug 8, 2016

The limit definition at a point is that the derivative of #f(x)# at #x=a# is:

#f'(a)=lim_(xrarra)(f(x)-f(a))/(x-a)#

So here, where #f(x)=3/(x+1)# and #a=2#:

#f'(2)=lim_(xrarr2)(3/(x+1)-3/(2+1))/(x-2)#

#f'(2)=lim_(xrarr2)(3/(x+1)-1)/(x-2)#

#f'(2)=lim_(xrarr2)(3/(x+1)-(x+1)/(x+1))/(x-2)#

#f'(2)=lim_(xrarr2)((3-(x+1))/(x+1))/(x-2)#

#f'(2)=lim_(xrarr2)(3-(x+1))/(x+1)(1/(x-2))#

#f'(2)=lim_(xrarr2)(-x+2)/(x+1)(1/(x-2))#

#f'(2)=lim_(xrarr2)(-(x-2))/(x+1)(1/(x-2))#

#f'(2)=lim_(xrarr2)(-1)/(x+1)#

#f'(2)=(-1)/(2+1)#

#f'(2)=-1/3#

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