How do I solve this Lim x--> + ∞ { ( √( x2 + 4x + 1) ) - x } using the Chain Rule?

Question

Hello ! I've managed to solve for the given equation's limit using the standard concepts involving derivatives, but I would like to find out how to solve it using the chain rule because it is apparently more convenient. Is this possible?

Answer 1

Hello, I am not sure how you managed to solve this limit using derivatives without using the chain rule. If you took a derivative, you had to use the chain rule (maybe without realizing it)

Also, you don't need to take derivatives or even use L'hopital's rule for this limit...just some algebra.

To save me from a lot of extra typing, let us agree to let:

1. F = √(x²+4x+1)
2. and G = x

With this then your limit looks like this:

1. Lim{x-->∞} F - G which does have the form of ∞-∞
2. However, when you have a indeterminate form like ∞-∞ you can often simplify things by rewriting the expression as a fraction.

Here is what I mean:

1. We can multiply by a form of one without changing the value of the expression. The form we will use is to multiply by (F+G)/(F+G)
2. This gives us (F-G)(F+G) for the numerator which simplifies to F²-G² = 4x+1
3. The denominator is F + G = √(x²+4x+1) + x

Note that we can factor out an x from both expressions like this:

1. 4x+1 = 4x+x/x = x(4+1/x) for the numerator
2. √(x²+4x+1) + x = √(x²+4x²/x + x²/x²) + x =
3. √[x²(1 + 4/x + 1/x²)] + x =
4. x√(1 + 4/x + 1/x²) + x =
5. x(√(1 + 4/x + 1/x²)+1) for the denominator
6. The x in the numerator reduces the x in the denominator to give us the final expression:

1. Lim{x-->∞} (4+1/x) / (√(1 + 4/x + 1/x²) + 1) which simplifies into 4/2 = 2 which is the limit sought without using derivatives or l'Hopitals rule or the chain rule.

Let me know if this was helpful or if you need a follow up...I could be more explicit with the algebra if you need it, but practicing your algebra skills will make you a much better calculus student.