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Evaluate the limit, if it exists.
$ \displaystyle \lim_{h \to 0}\frac{\sqrt{9 + h}-3}{h} $
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03:09
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 3
Calculating Limits Using the Limit Laws
Limits
Derivatives
Campbell University
Harvey Mudd College
Baylor University
University of Nottingham
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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03:34
So in this problem were asked to determine the limits His age goes to zero of the squirt of nine plus H minus three. All over. H. All right. Notice that we can multiply this By the squirt of nine plus H plus three over itself. Because we can always multiply limit by one and still have the limit, right? Because one times anything is itself. So then this becomes the limit as h goes to zero of well nine plus the screwed at nine H minus three times. To scrutinize Jose plus three is just the first term squared minus the second term squared. So that means I end up with nine plus H- Well three square it is nine Over eight times the square root of nine plus H plus three. Okay, So this is the limit as h goes to zero of well 9 -9. That's those cancel out and H over the H then cancels out. Doesn't it? Let me let me do this one step at a time. So we're not getting confused here. So it's age over eight times The skirt of nine plus H. Was three. And now the H is canceled. So I'm left with limit As a church goes to zero of one over The square to nine plus H. Last three. Well as H goes to zero. Look at this the limit as H goes to zero. The screw nine plus H is simply what the screw to nine which is three. So this becomes 1/3 plus three which is 1/6. So our answer for this limit Is 1/6.
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