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Evaluate the limit, if it exists.
$ \displaystyle \lim_{h \to 0}\frac{(x + h)^3 - x^3}{h} $
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02:18
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 3
Calculating Limits Using the Limit Laws
Limits
Derivatives
Oregon State University
Harvey Mudd College
Baylor University
Idaho State University
Lectures
04:40
In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
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So here were asked to evaluate specific limit. The limit as h approaches zero of X plus H cube minus X cubed divided by H. So let's first expand the numerator. So manly we're doing a binomial expansion here. So this would just be X cubed plus three X squared H plus three X H squared plus H cubed minus X cubed divided by H. So this is equivalent to the limit. As H approaches zero, the X keeps canceled out. And we also can factor out an H. In this case. So we have a church of three X squared Plus three x. H plus H squared divided by H. And we can cancel out the H is from the numerator and denominator. And we're left with the limit as h approaches zero of three x squared plus three X H plus H squared. We don't have any indeterminant form anymore. So we can directly evaluate so H zero and zero. So this would just be equivalent to three X squared. So this was giving the general limit form of the derivative of execute and we could also find the derivative of X cubed mainly by taking the derivative of X cubed which is just equivalent to three X squared, which agrees with this. Alternatively we can also apply lawful rule where we would take the derivative of the numerator and the derivative the denominator in order to find the limit. But that would be slightly more complicated in this case and we can just use the limit definition of the derivative to evaluate the derivative. And this is our final answer
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