I have an expression such as the following


which describes a physical system that I want to find the limit as $x$ goes to 0. If I simply ask Mathematica to take the limit, I get DirectedInfinity with some argument.

However, if I plug in some numbers and instead tell Mathematica to take the limit in that case, I get a finite result. One example of this is that if I use $a=10$, $b=5$, and $c=3$, I get that the limit is $$\frac{1}{2}+\frac{d}{7}$$

I would like to have an expression for the limit generically when $a$, $b$, $c$, $d$, and $x$ are all positive real numbers.


closed as off-topic by Daniel Lichtblau, Feyre, corey979, user31159, Michael E2 Nov 2 '16 at 22:00

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Bulding on my comment.

Limit can work with Assumptions (see the section Options within the documentation).

In this case you should write:

Limit[1/(2 a x) (a b + a^2 c + b x - a c x + 
    2 a d x - (a - x) Sqrt[b^2 + 2 a b c + a^2 c^2 + 4 a c d x]), 
 x -> 0, Assumptions -> {a > 0, b > 0, c > 0, d > 0, x > 0}]

With result:

(*(b (b + a (c + d)))/(a (b + a c))*)

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