# Find the limit of (sqrt(1+2x) - sqrt(1+3x))/(x + 2x^2) as x->0

Gold Member
I'm having a lot of trouble making any progress on this limit. If someone could give me a direction to get started I would appreciate it.

$$\lim_{x{\rightarrow}0}\frac{\sqrt{1+2x}-\sqrt{1+3x}}{x+2x^2}$$

Use L'Hopitals Rule.

StatusX
Homework Helper
Or you could multiply by the conjugate of the numerator. (ie, sqrt(1+2x)+sqrt(1+3x)).

JasonRox
Homework Helper
Gold Member
Go with StatusX's advice if L'Hopitals Rule wasn't taught yet because then it wouldn't be appropriate.

If you haven't learned L'Hopitals Rule, I recommend to learn it and use it to check your answer

Also, you can always test it numerically. I do that sometimes just to be 100% certain. I would try something like x=0.01, 0.001 and 0.0001.

As stated above, L'Hopital's rule, or, if that's not allowed, rationalize the numerator.