Intermediate value theorem with an interval

Question

$a,b > 0$ and $c<d$

Show that $g(x) = \displaystyle \frac{a}{(x-c)^4} + \frac{b}{(x-d)^7}=0$ has at least a solution in $(c,d)$

I defined

$\lim\limits_{x\searrow c}g(x)=+\infty$

$\lim\limits_{x\nearrow d}g(x)=-\infty$

I don't know if the limits are just enough for the proof, or is there something else?

Answer 1

The limits are almost enough. You just need to finish up by saying that the limits imply the existence of positive/negative points, and the continuity of the function on the interval $(c, d)$ implies a zero is contained in the interval (by the Intermediate Value Theorem).