Evaluate the limit, if it exists. (If it does not exist, enter NONE).
lim h→0 (3 + h)3 − 27
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h
To evaluate the limit of this expression, let's simplify it step by step.
First, let's substitute h = 0 into the expression:
lim h→0 (3 + h)^3 - 27
_______________
h
This simplifies to:
(3 + 0)^3 - 27
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0
Simplifying further:
(3)^3 - 27
_________
0
= 27 - 27
_______
0
= 0
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Since we obtained a finite value of 0, the limit exists, and the value of the limit is 0.
To evaluate the limit, we can substitute the value of h equal to 0 into the expression and simplify. Let's go step by step:
1. Substitute h = 0 into the expression:
(3 + 0)^3 - 27
Simplify:
3^3 - 27
2. Evaluate the expression:
3^3 = 27, so we have:
27 - 27 = 0
3. Since the numerator is 0, we need to check the denominator. If the denominator is also 0, then the limit does not exist. Let's check the denominator:
h
4. Substitute h = 0 into the denominator:
0
5. The denominator is also 0, which means the limit does not exist.
Therefore, the limit of the given expression as h approaches 0 does not exist.