# Evaluate the limit #lim_(h rarr 0) {(x-h)^3-x^3}/h#?

# lim_(h rarr 0) {(x-h)^3-x^3}/h = -3x^2 #

We seek:

Therefore:

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To evaluate the limit ( \lim_{h \to 0} \frac{(x - h)^3 - x^3}{h} ), we can expand the numerator using the binomial expansion formula:

[ (x - h)^3 = x^3 - 3x^2h + 3xh^2 - h^3 ]

Then, subtract ( x^3 ) from both sides to simplify:

[ (x - h)^3 - x^3 = - 3x^2h + 3xh^2 - h^3 ]

Now, substitute this expression into the original limit:

[ \lim_{h \to 0} \frac{(-3x^2h + 3xh^2 - h^3)}{h} ]

Factor out an ( h ) from the numerator:

[ \lim_{h \to 0} \frac{h(-3x^2 + 3xh - h^2)}{h} ]

Cancel out ( h ) from the numerator and denominator:

[ \lim_{h \to 0} -3x^2 + 3xh - h^2 ]

Now, plug in ( h = 0 ) into the expression:

[ -3x^2 + 3x(0) - (0)^2 = -3x^2 ]

Therefore, the limit ( \lim_{h \to 0} \frac{(x - h)^3 - x^3}{h} ) evaluates to ( -3x^2 ).

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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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