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Problem 33 Medium Difficulty

(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.

Answer

a. $\frac{2}{3}$
b. $$\begin{array}{|c|c|}
\hline x & f(x) \\
\hline-0.001 & 0.6661663 \\
-0.0001 & 0.6666167 \\
-0.00001 & 0.6666617 \\
-0.000001 & 0.6666662 \\
0.000001 & 0.6666672 \\
0.00001 & 0.6666717 \\
0.0001 & 0.6667167 \\
0.001 & 0.6671663 \\
\hline
\end{array}$$
c. $\frac{2}{3}$

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Video Transcript

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