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(a) Use a graph of $$ f(x) = \frac{\sqrt{3 + x} - \sqrt{3}}{x} $$to estimate the value of $ \displaystyle \lim_{x \to 0}f(x) $ to two decimal places.

(b) Use a table of values of $ f(x) $ to estimate the limit to four decimal places.

(c) Use the Limit Laws to find the exact value of the limit.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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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(a) Use a graph of$$f(…

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this problem Number thirty four was this tour Calculus Safe Edition, section two point three Pretty use a graft. Half of X equals this corbett of three plus six minus a square to three wrecks to estimate the value How the limit is X purchase here over this function to do it to test in places simply plant dysfunction and as is foreign and we see that it approach is very close to zero point two nine. So this limit and party we will apart that parks me for it to be a hero. Two nine party. He's a table of values of F Testament limited for distant places. Oh, wait. Use the table to plot this function very close to zero and we see that closer than we got two zero from the left and the right, of course, are we approximated the function value to be point two eight. What happened? Two. Four decimal places and the letter there knew it. Improved estimate party use element lab laws to find the exact value of elements. So where you will really write this as a limit on the next page. Limited expert zero of screwed of three sex Explain over minus the square root of And we will take the next step of rationalizing the numerator. Yeah, by multiplying by square root and Papa's X plus the square root of three to the temple about him, which could thiss three plus sex minus three in the numerator divided by X Time's a quality square root of the quantity three point six, plus a spurt of three we see here at the three minus three, it cancels out to zero, and then we're left with her. One acts in the top, which can cancel with the one X and that there no need her leaving us with no limit his expressions. He around, uh, one over in the square root of the quantity three parsecs plus square root of three. And if we evaluate us at X equals zero, that should give us and exact value one over this herd of three plus a screw to three or one over two times the square root of three. And this is our exact solution. This product are this questioned is equal to approximately on zero point two eight eight seven, which is consistent with our yes, um, it's, um, ists from parts A and B

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