Limit

Calculus Level 2

$\lim_{x\to2} \dfrac{\sqrt{1+\sqrt{2+x}} - \sqrt3}{x-2}$

If the value of the limit above is equal to $\dfrac 1{a\sqrt b}$ , where $a$ and $b$ are integers with $b$ square-free, find $\dfrac{2016}{a\times b}$ .

The answer is 84.

2 solutions

Chew-Seong Cheong
Jan 24, 2016

Let $f(x) = \sqrt{1+\sqrt{2+x}} - \sqrt{3}$ and $g(x) = x -2$ . We note that $\displaystyle \lim_{x \to 2} \frac{f(x)}{g(x)} = \frac{0}{0}$ , therefore we can use L'Hôpital's rule.

$\begin{aligned} \lim_{x \to 2} \frac{\sqrt{1+\sqrt{2+x}} - \sqrt{3}}{x-2} & = \lim_{x \to 2} \frac{f(x)}{g(x)} \\ & = \lim_{x \to 2} \frac{f'(x)}{g'(x)} \\ & = \lim_{x \to 2} \frac{\frac{1}{2}\left(1+\sqrt{2+x} \right)^{-\frac{1}{2}} \dot{} \frac{1}{2} \left(2+x \right)^{-\frac{1}{2}} \dot{} 1}{1} \\ & = \lim_{x \to 2} \frac{1}{4 \sqrt{1+\sqrt{2+x}} \dot{} \sqrt{2+x}} \\ & = \frac{1}{8\sqrt{3}} \end{aligned}$

$\Rightarrow \dfrac{2016}{ab} = \dfrac{2016}{8\times 3} = \boxed{84}$

From the definition of the derivative, the given limit is simply $f'(2)$ .

Jon Haussmann - 5 years, 4 months ago

You are right.

Chew-Seong Cheong - 5 years, 4 months ago

If proof is not required, then calculator allows an easy guess.

Lu Chee Ket - 5 years, 4 months ago

Please follow the community. Just do your Algebra. Using an Excel speadsheet, which I am very good at, will almost never fail to find the solution but it is not wanted here.

Chew-Seong Cheong - 5 years, 4 months ago

I have reduced my will to posting solutions. Instead, I entered at Reply when I wanted to say few opinions. What you tell is expected by me.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – Brilliant is a platform to promote Mathematics and not use of calculator. If the members just want the answer, we can use Wolfram Alpha saving all the time punching in a calculator. We want the math behind it. You may feel that you have achieved something by punching calculator and get the right answers but not most of us.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Obtaining answers should look for easiest ways. Nevertheless, proving the mathematics ought to go through all that you have mentioned.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – That is what I said. Wolfram Alpha is definitely easier than a calculator. Easiest way in terms of mathematics not tools. We should look at all problems in Brilliant with an unwritten rule that get the solution without the use of computation tools unless specified. That is why some of the problems, the authors specified that the use of calculator may be necessary for the final computation. Numerical method and graphing are the last resort here.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Questions in Brilliants are not the same as usual examinations. We answer closer to whether we agree or not rather than to force to give an answer when we don't know how to solve them. With plenty of time and opportunity to study before answering, the important thing is to find good way to solve. Wolfram Alpha is not the source of everything and its application does not include all sorts of human interpretations. Questions in Brilliant should not restrict onto ways to solve but they ought to be good questions that are mind provoking.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – Has any one of the solutions get upvoted? If not why do you think it is so. Of course, every member is free to do anything they like with their solutions. Just trying to say what a normal member should do. I guess you will not understand that you are not doing math but doing calculator. Doing math means using math theorems. I will not comment on the matter anymore.

Chew-Seong Cheong - 5 years, 4 months ago

@Lu Chee Ket – Don't get me wrong. I was not saying you must use Wolfram Alpha. I was saying solving problems in Brilliant is not just about getting the answers. Because every one can get the most of answers easily through Wolfram Alpha. It is just a super calculator. Solving Brilliant problems is about how to use math theorem. That is why there are so many Wiki's in Brilliant teaching about theorems. Else we will need only one wiki -- how to use a calculator properly.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Write computer program to solve is another way of applying calculator. The making of software calculator is itself an actualization of mathematical theorem.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – Another way of applying calculator? We are saying don't use calculator. We are saying apply theorems in solving the math problems not how you punch a calculator. There are a lot of Computer Science problems for you to do that.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Let's not to think that I am answering to this question. As another topic, implementing mathematical theorem into software program to evaluate can sometime help to solve certain questions which cannot be solved easily just by hand, pen and papers, as some theorem states explicitly that certain thing is proven to be an impossible.Yet, iteration or numerical method allows a continuation onto the study into depth.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – You are right. For other topics but not this problem. When it applies then you comment. Not in this problem. Frankly, before I wrote the solution, I solve it with Wolfram Alpha or a spreadsheet just to make sure that it is correct. Then I apply theorem. In this case it is L'Hôpital's rule. So that other members can learn. But the solution provided by Jon Haussmann is simpler. I didn't see it because, I have been relying on software too much.

Frankly, I am very good with spreadsheet. We can calculate f(x) here for x = 1.9, 1.99, 1.999, 1.9999 .... and see it converges. We can plot graphs to find roots, maxima and minima. I solved many problems like that but I don't report them. Other members are not interested in the computed data. But after knowing Wolfram Alpha, I don't need to use spreadsheet.

That is why many of the answers needed in Brilliant are in integer form instead of decimal because numerical methods or batching are available for many problems.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – I obtained a guessed value for $a^2 b$ using calculator. Then I determined its prime factors and solved eventually.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – When it is a calculus problem let us only comment on calculus.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Substitute $x$ = 2.001 for example, enable us to get $a^2 b \approx$ 192.0320018, treated as 192 is $2^6 \times 3.$ This was actually how I solved this question.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – I am not interested because that is not calculus. Calculus need your $\delta (0.001) \to 0$ .

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – Well, x = 2.00000001 gives 192.0000003199999999666666667299 instead with a calculator with more significant figures.

Lu Chee Ket - 5 years, 4 months ago

@Lu Chee Ket – Well, still not calculus. You are not interested in what others are interested in only what you are interested. Anyway, I won't waste more time convincing you. You just love numerical methods.

Chew-Seong Cheong - 5 years, 4 months ago

@Chew-Seong Cheong – I believe this is a part of fundamental concept behind Calculus to my understanding.

Lu Chee Ket - 5 years, 4 months ago

Deepak Kumar
Jan 23, 2016

Hint:Use rationalization, to combat 0/0 form here and it is easy to observe that the factor of (x-2) from ddenominator will be eliminated after using rationalisation twice which makes it easy to calculate without pen and paper with little oobservation.