Calculating limit $\lim\limits_{x\to\infty}\frac{3x^2-\frac{3}{x^2+1}-4f'(x)}{f(x)}$ for an unknown function.

Question

Given that $f(x)$ is a continuous function and satisfies $f'(x)>0$ on $(-\infty,\infty)$ and $f''(x)=2 \forall x \in(0,\infty)$.

We need to find the limit

$$\lim_{x\to\infty}\frac{3x^2-\frac{3}{x^2+1}-4f'(x)}{f(x)}$$

Now the numerator is tending to infinity so denominator must also go to infinity else limit won't exist.So I tried the L'Hospitals rule and it became$$\lim_{x\to\infty}\frac{6x+\frac{6x}{(x^2+1)^2}-4f''(x)}{f'(x)}$$The numerator is still infinity so once again applying L'Hospitals rule (assuming denominator must still be infinity) we get

$$\lim_{x\to\infty}\frac{6+\frac{6(x^2+1)^2-6x×2(x^2+1)×2x}{(x^2+1)^4}+0}{f''(x)}$$

Now putting $f''(x)=2$ we get

$$3+\lim_{x\to\infty}\frac{3(x^2+1)^2-12x^2(x^2+1)^2}{(x^2+1)^4}$$

Collecting the coefficients of $x^4$ from numerator and denominator we get the limit to be$3-9=-6$ but the answer is not -6.

Is applying LHospital wrong?Help.
Thanks.

Answer 1

The two given conditions mean $\;\lim\limits_{x\to\infty}f(x)=\infty\;$ (why?), and then, from your work, we reached

$$\lim_{x\to\infty}\frac{6+\frac{6(x^2+1)^2-6x×2(x^2+1)×2x}{(x^2+1)^4}+0}{f''(x)}=\frac{6+0+0}2=3$$

I don't really understand what you did after the above part in your question...but the middle summand in the numerator above is

$$\frac{6(x^2+1)-24x^2}{(x^2+1)^3}\xrightarrow[x\to\infty]{}0$$

Answer 2

Since $f''(x)=2~~for ~~x>0$, Therefore $f$ has the form $f(x)=x^2+bx +c~~~for~~~x>0$

Since the limit is at $+\infty$ it suffices to consider $x>0$ $$\lim_{x\to\infty}\frac{3x^2-\frac{3}{x^2+1}-4f'(x)}{f(x)} = \lim_{x\to\infty}\frac{3x^2-\frac{3}{x^2+1}-8x-4b}{x^2+bx +c} =3$$

Answer 3

The issue is with the largest power of $x$. At the numerator is $4$, at the denominator is $8$ (because $(x^2+1)^4$). So the last limit is $0$

Answer 4

From $f''(x)=2$, we know that $f$ is a quadratic function with leading term $x^2$. There will be no cancellation of $3x^2$ at the numerator, and by ignoring low order terms, the expression simplifies to

$$\frac{3x^2}{x^2}.$$

Answer 5

The condition $f''(x) =2$ makes the problem far simpler because then we can use integration to get $f(x) =x^2+bx+c$ and perform the limit evaluation easily. The answer however remains the same if we are given the weaker hypothesis that $f''(x) \to 2$ as $x\to\infty$ and I prove this below.

First we observe that derivative $f'$ is positive and hence $f$ is strictly increasing. Therefore $f(x) $ tends to a limit or to $\infty$ as $x\to\infty$. By similar argument $f'(x) $ also tends to a limit or to $\infty$. If $f(x) $ tends to a limit then $f(x+1)-f(x)=f'(c)$ (via mean value theorem) tends to $0$. This is a contradiction as derivative $f'$ is positive and strictly increasing for large $x$ so it can't tend to $0$. Thus it follows that $f(x) \to\infty$ as $x\to\infty$. Similarly $f'(x) \to \infty$ because $f''(x) \to 2$.

Thus the desired limit is equal to the limit of $$\frac{3x^2-4f'(x)}{f(x)}$$ And since denominator tends to $\infty $ we can try L'Hospital's Rule to get the ratio $$\frac{6x-4f''(x)}{f'(x)}$$ whose limit is same as that of $6x/f'(x)$. Applying L'Hospital's Rule once again we get the ratio $6/f''(x)$ which tends to $3$ and therefore the original limit is also $3$.