Evaluate this limit in terms of f

Question

I want to evaluate the following limit:

$$\lim_{d\to x} \dfrac{\dfrac{2x}{f'(x)}+f(x)-f(d)-\dfrac{x^2-d^2}{f(x)-f(d)}}{2\left(\dfrac{d-x}{f(x)-f(d)}+\dfrac{1}{f'(x)}\right)}$$

I tried L'hopital's rule but it just keeps getting worse and worse.

I got this limit by wondering about circles fitting on a curve at a point $(x,f(x))$ and this limit is the $x$ coordinate of the circles center, in terms of a dummy point $d$.

Do you know the limit-based definition of $f'(d)$? Can you write it out for us? Do you see anything like that anywhere in your numerator or denominator? — John Hughes, Dec 14 '14 at 14:54
May be, you could start developing $f(d)$ as a Taylor series built at $d=x$. — Claude Leibovici, Dec 14 '14 at 14:58
@JohnHughes I've tried that, but it yields a 0/0 and l'hopital makes it even messier and still a 0/0 if I've done it correctly. — Faraz Masroor, Dec 14 '14 at 20:02

Sam Dittmer · Answer 1 · 2014-12-17T09:34:58.393

This isn't pretty, but it works:

Multiply the top and bottom by $f'(x)(f(x)-f(d))$ to clear denominators, giving: $$ \lim_{d \to x} \frac{2x(f(x)-f(d))+f'(x)(f(x)-f(d))^2-f'(x)(x^2-d^2)}{2(f(x)-f(d)-f'(x)(x-d))} $$ By adding $(x-d)^2f'(x)-(x-d)^2f'(x)$ to the numerator, this becomes $$ \lim_{d \to x} \frac{2x(f(x)-f(d))-2xf'(x)(x-d) + f'(x)(f(x)-f(d))^2+f'(x)(x-d)^2}{2(f(x)-f(d)-f'(x)(x-d))} $$ $$ = x + \lim_{d \to x} \frac{f'(x)(f(x)-f(d))^2+f'(x)(x-d)^2}{2(f(x)-f(d)-f'(x)(x-d))} $$

Now use L'Hopital's Rule, differentiating with respect to $d$, to give: $$ =x+ \lim_{d \to x} \frac{-2f'(x)f'(d)(f(x)-f(d))-2f'(x)(x-d)}{2f'(x)-2f'(d)} $$ This is still an indeterminate form, so cancel the $2$'s, and use L'Hopital's Rule again: $$ =x + \lim_{d \to x} \frac{f'(x)f'(d)^2-f'(x)f''(d)(f(x)-f(d)) + f'(x)}{-f''(d)} $$ $$ =x - \lim_{d \to x} \frac{f'(x)f'(d)^2+f'(x)}{f''(d)} $$ $$ =x - \frac{f'(x)+(f'(x))^3}{f''(x)} $$ As long as $f''(x) \neq 0$.

Evaluate this limit in terms of f

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