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(a) Estimate the value of $$ \lim_{x \to 0}\frac{x}{\sqrt{1 + 3x} - 1} $$by graphing the function $ f(x) = x/(\sqrt{1 + 3x} - 1) $.

(b) Make a table of values of $ f(x) $ for $ x $ close to 0 and guess the value of the limit.

(c) Use the Limit Laws to prove that your guess is correct.

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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

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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this problem. Number thirty three of the sewer calculus eighth edition Section two point three Party has to meet. The value of the limit is ancient experts. Zero of extroverted pine, the square root of the quantity on plus three x minus one. My graphing The function Evan X equals X, divided by the quantity square root of the quantity. One mystery X minus one. If we plant dysfunction is in any party utility, see that it resembles a straight line and it approaches about point six or repeating from the left and from the right. And so we would estimate this limit at this point to be a power point six repeating or about two thirds from the graph party. Make a table of values of half for X close to zero and guess the valuable element in our spreadsheet. We used these values to graph the plan. Bali is at great cost zero and is he Considine As we approach from the right hand from the left, the values seem to approach the value of zero point six. Repeating S o. The table table of values method also confirms our estimate of Lim being equal to two thirds party. Use a limit laws to prove that your guess is correct. So he will ah rewrite to limit on the next page Lim and sex britches hero tied up by the quantity screwed of one history X minus one and what we're going to do as we're going to use a country, it of the denominator to rationalize it. That will be the quantity on P three x square plus one. Want to play to the top and the bottom and we see that you get he rationalized nominator one plus three x minus one and then the numerator x times screwed of the quantity on those three acts, plus one here one and eight of one. Cancel X and X or cancel as they as it is shown right there. And then we have a much simpler limit as experts zero the square root of the quantity one post reacts plus one divided by three has experts zero. This approach is one plus one over three, which is equal to two thirds, and this value agrees various parts A and B

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