How to calculate this limit? f:[0,1]#->RR#,f(x)=x#sqrt(1-x^2)# Calculate #lim_(x->0)1/x^2int_0^xf(t)dt#

Answer 1

#lim_(xrarr0)1/x^2int_0^xf(t)dt=1/2#

#lim_(xrarr0)1/x^2int_0^xtsqrt(1-t^2)dt#
As #x# approaches #0#, note that there are two things happening: #1/x^2# becomes #1/0# and the integral approaches #int_0^0f(t)dt#, which is also #0#.
Therefore the limit is in the form #0/0#, which means that l'Hopital's rule applies.

We can then treat it as:

#lim_(xrarr0)(int_0^xtsqrt(1-t^2)dt)/x^2#

And then using l'Hopital's rule:

#=lim_(xrarr0)(d/dxint_0^xtsqrt(1-t^2)dt)/(d/dxx^2)#
The derivative of the numerator can be found through the Second Fundamental Theorem of Calculus. The derivative of the denominator is #2x#:
#=lim_(xrarr0)(xsqrt(1-x^2))/(2x)=lim_(xrarr0)sqrt(1-x^2)/2=1/2#
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Answer 2

To calculate the limit, first integrate the function ( f(t) ) from 0 to ( x ) and then divide the result by ( x^2 ). So, the steps are:

  1. Integrate ( f(t) = x\sqrt{1-x^2} ) with respect to ( t ) from 0 to ( x ).
  2. Divide the result by ( x^2 ).
  3. Finally, take the limit as ( x ) approaches 0.

Now, let's perform the calculation.

  1. Integrate ( f(t) = x\sqrt{1-x^2} ) with respect to ( t ) from 0 to ( x ):

[ \int_{0}^{x} f(t) dt = \int_{0}^{x} t\sqrt{1-t^2} dt ]

  1. Perform the integration:

[ = \frac{-1}{3}(1-t^2)^{\frac{3}{2}} \Bigg|_{0}^{x} ]

[ = \frac{-1}{3}(1-x^2)^{\frac{3}{2}} - \frac{-1}{3}(1-0^2)^{\frac{3}{2}} ]

[ = \frac{-1}{3}(1-x^2)^{\frac{3}{2}} - \frac{-1}{3} ]

[ = \frac{-1}{3}(1-x^2)^{\frac{3}{2}} + \frac{1}{3} ]

  1. Divide the result by ( x^2 ):

[ \frac{1}{x^2} \int_{0}^{x} f(t) dt = \frac{1}{x^2} \left( \frac{-1}{3}(1-x^2)^{\frac{3}{2}} + \frac{1}{3} \right) ]

[ = \frac{-1}{3x^2}(1-x^2)^{\frac{3}{2}} + \frac{1}{3x^2} ]

  1. Finally, take the limit as ( x ) approaches 0:

[ \lim_{x \to 0} \frac{1}{x^2} \int_{0}^{x} f(t) dt = \lim_{x \to 0} \left( \frac{-1}{3x^2}(1-x^2)^{\frac{3}{2}} + \frac{1}{3x^2} \right) ]

[ = \frac{-1}{3}(1-0)^{\frac{3}{2}} + \frac{1}{3} ]

[ = -\frac{1}{3} + \frac{1}{3} ]

[ = 0 ]

So, the limit as ( x ) approaches 0 of ( \frac{1}{x^2} \int_{0}^{x} f(t) dt ) is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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