How do you find the lim & definite integrals ? for :{lim as x approaches 0 (1/x^3) * integrals (t^2/t^4+1)dt} intervals{0,x}

1 Answer
Ultrilliam
May 14, 2018

#= 1/3#

Explanation:

#lim_( x to 0) ( int_0^x (t^2/(t^4+1))dt)/x^3#
This is #0/0# indeterminate and made for L'Hopital's rule -- the numerator can be unraveled swiftly by differentiating.

Applying L'H:

#= lim_( x to 0) ( d/dx int_0^x (t^2/(t^4+1))dt)/(d/dx (x^3 ) )#

#= lim_( x to 0) ( x^2/(x^4+1))/(3 x^2)#

#=1/3 lim_( x to 0) 1/(x^4+1) = 1/3#

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