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How do you find the limit of #(x-tanx)/x^3# as x approaches 0?

Answer 1
Isabella AckermanIsabella Ackerman

In this way:

#lim_(xrarr0)(x-tanx)/x^3=#Hospital#=lim_(xrarr0)(1-1-tan^2x)/(3x^2)=#
#=lim_(xrarr0)-sin^2x/(3x^2cos^2x)=lim_(xrarr0)-(sinx/x)^2*1/(3cos^2x)=#
#=-1^2*1/(3*1)=-1/3#.
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Answer 2
Christopher AgrestaChristopher Agresta

To find the limit of (x - tan(x))/x^3 as x approaches 0, we can use L'Hôpital's Rule. Taking the derivative of the numerator and denominator separately, we get (1 - sec^2(x))/3x^2. Substituting x = 0 into this expression, we get (1 - 1)/0, which is undefined. Therefore, the limit does not exist.

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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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