Here is the question,

I calculated it and got square root of 5 as an answer. Wolfram Alpha has the same solution.

But the actual answer key says it's 3/2!(not factorial, lol) By what kind of calculation is such answer even possible? Please help me out :(

I think it is 3/2. You have (A-B). If you take (A-B)(A+B)/(A+B) then simplify the top then the bottom is 2x when x is large.

(Sqrt[x^2 + 4 x + 5] - Sqrt[x^2 + x])*(Sqrt[x^2 + 4 x + 5] +

Sqrt[x^2 + x])/(Sqrt[x^2 + 4 x + 5] +

Sqrt[x^2 + x]) ~ (5 + 3x)/(2x) -> 3/2

One more thing. I calculated the limit as x goes to infinity. If the limit is x goes to zero then the answer is square root of five which is pretty simple since it is continuous.

@Jack Merrin Thanks! I guess it's just a typo from examiners then.

Jack is correct.

x ---> 0, then L = sqrt(5)

x ---> inf, then L= 3/2

My guess is that there was a typo when the question was made up and that the intention was inf and not 0.

That confirmed my suspicion. I knew something was wrong since the answer key had a different answer from the GOD wolfram alpha itself.

This illustrates another cool fact about the (inf - inf) indeterminate form. When you subtract the square roots of 2 quadratics, as x---> inf, you can make the limit equal to any number you want by just choosing the appropriate co-efficients.

https://drive.google.com/file/d/13-pzP7NABjiNpEat1f4DKikvxi6WDLck/view?usp=sharing Seethe referred PDF to have an idea of the question.