Linked Questions

6
votes
5answers
3k views

Limit of the derivative of a function [duplicate]

$f(x)$ is a differentiable function on the real line such that $ \lim_{x\to \infty } f(x) =1 $ and $ \lim_{x\to \infty } f'(x) = s $ .Then $s$ should be $0$ $s$ need not be $0$ but $|s| < 1$ $s &...
4
votes
5answers
233 views

Prove that $\displaystyle \lim_{x \to \infty} f'(x) = 0$ [duplicate]

Prove that if $\displaystyle \lim_{x \to \infty} f(x)$ and $\displaystyle \lim_{x \to \infty} f'(x)$ are both real numbers, then $\displaystyle \lim_{x \to \infty} f'(x) = 0$. Attempt Intuitively ...
5
votes
2answers
374 views

Proving the limit at $\infty$ of the derivative $f'$ is $0$ if it and the limit of the function $f$ exist. [duplicate]

Suppose that $f$ is differentiable for all $x$, and that $\lim_{x\to \infty} f(x)$ exists. Prove that if $\lim_{x\to \infty} f′(x)$ exists, then $\lim_{x\to \infty} f′(x) = 0$, and also, give an ...
5
votes
2answers
172 views

Show that if $\lim\limits_{x \to \infty} f(x)$ exists and $f''$ is bounded, then $\lim\limits_{x \to \infty} f'(x)=0$. [duplicate]

I'm trying to answer the following exercise: Suppose that $f$ is twice differentiable on $[0,+\infty)$. Show that if $\lim\limits_{x \to \infty} f(x)$ exists and $f''$ is bounded, then $\lim\...
1
vote
3answers
515 views

If Limit of function and derivative exist, then limit of derivative is 0 [duplicate]

Any hints for this question , My attempt; Say $f(x):0$$\rightarrow$$\mathbb{R}$ The by MVT, there exists a $c$$\in$$(0,\infty)$ , such that; $f'(c)=$$\frac{f(x)-f(0)}{x-0}$ but im not sure about this ...
4
votes
2answers
90 views

$\underset{x\rightarrow\infty}{\lim}\frac{f(x)}{x}=0$ Implies $\underset{x\rightarrow\infty}{f'(x)}=0$ [duplicate]

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuously differentiable function such that $\underset{x\rightarrow\infty}{\lim}\frac{f(x)}{x}=0$ and suppose $\underset{x\rightarrow\infty}{f'(x)}$ ...
0
votes
1answer
275 views

$\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = L$ Prove $\mathop {\lim }\limits_{x \to \infty } f'\left( x \right) = 0$ [duplicate]

Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a continuously differentiable function such that $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = L~~and~~~\mathop {\lim }\limits_{x \...
-1
votes
1answer
145 views

Does $f(x)$ converge when $x$ goes to infinity if $f'(x)$ goes to $0$ when $x$ goes to infinity? [duplicate]

Clarification: Does $\lim\limits_{x \to \infty}f'(x)=0$ as $x$ approaches infinity mean $\lim\limits_{x \to \infty} f(x)$ exists in the extended real numbers $[-\infty,\infty]$? I was thinking a lot ...
1
vote
1answer
105 views

If $f$ and $f'$ have a limit then is it equal to zero for $f'$? [duplicate]

I have this question : Let $f:\mathbb{R}\rightarrow \mathbb{R}$ , and $f\in C^1(\mathbb{R})$ such that $$\begin{cases}\lim_{t\to\infty} f(t)=l_1\\ \lim_{t\to\infty}f'(t)=l_2\end{cases}$$ $l_1,\, ...
2
votes
2answers
88 views

Limit of $y$ if limit of $y+y'$ goes to $0$? [duplicate]

Let $y$ be a differentiable function on $\mathbb{R}$. If $$\lim_{x\rightarrow \infty}(y(x)+y'(x))=0$$ Then how does one show that $$\lim_{x\rightarrow \infty}y(x)=0$$ I'd appreciate some help on this ...
2
votes
1answer
90 views
3
votes
0answers
114 views

finite $\lim_{x \rightarrow +\infty} f(x)$ and $\lim_{x \rightarrow +\infty} f'(x)$. Then $\lim_{x \rightarrow +\infty} f'(x) = 0$ [duplicate]

Let f be twice differentiable function on R with finite $\lim_{x \rightarrow +\infty} f(x)$ and $\lim_{x \rightarrow +\infty} f'(x)$. Then $\lim_{x \rightarrow +\infty} f'(x) = 0$ I don't know how ...
1
vote
1answer
46 views

If the limits at infinity of $f$ and $f'$ exist and are finite, then is $\lim_{x \rightarrow \infty} f'(x) = 0$? [duplicate]

Let $f$ be a continuously differentiable function on $\mathbb{R}$ such that both the limits $\lim_{x \rightarrow \infty} f(x)$ and $\lim_{x > \rightarrow \infty} f'(x)$ exist and are finite. Is $\...
0
votes
2answers
31 views

A differentiable function with itself and its derivative converge to constants, can we conclude its derivative converge to zero [duplicate]

I have been struggled with the following problem. Suppose that $f$ is differentiable on $(0, \infty)$. If we have $\lim_{x \rightarrow \infty} f(x) = c_1$ and $\lim_{x \rightarrow \infty} f'(x) = c_2$...
27
votes
6answers
13k views

If a function has a finite limit at infinity, does that imply its derivative goes to zero?

I've been thinking about this problem: Let $f: (a, +\infty) \to \mathbb{R}$ be a differentiable function such that $\lim\limits_{x \to +\infty} f(x) = L < \infty$. Then must it be the case that $\...

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