How do you evaluate the limit #((3+h)^3-27)/h# as h approaches #0#?

Answer 1

#lim_(hrarr0)((3+h)^3-27)/h=27#

The first step is to simplify #(3+h)^3=(h+3)^3#, either by multiplying out or by using the binomial theorem if you know it.
#(h+3)^3=(h+3)^2(h+3)#
#=(h^2+6h+9)(h+3)#
#=h^3+9h^2+27h+27#

Then:

#lim_(hrarr0)((3+h)^3-27)/h=lim_(hrarr0)((h^3+9h^2+27h+27)-27)/h#
#=lim_(hrarr0)(h^3+9h^2+27h)/h#
#=lim_(hrarr0)(h(h^2+9h+27))/h#
#=lim_(hrarr0)(h^2+9h+27)#
#=0^2+9(0)+27#
#=27#
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Answer 2

To evaluate the limit ((3+h)^3-27)/h as h approaches 0, we can simplify the expression by expanding (3+h)^3 using the binomial theorem. This gives us (27 + 27h + 9h^2 + h^3 - 27)/h. Simplifying further, we get (27h + 9h^2 + h^3)/h. Canceling out the h in the numerator and denominator, we are left with 27 + 9h + h^2. Now, as h approaches 0, the limit of this expression is 27.

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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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