Intereting Posts

Evaluation of $\sum_{x=0}^\infty e^{-x^2}$
Find shortest primary decomposition.
Perturbative solution to $x^3+x-1=0$
Equilateral triangle inscribed in a triangle
What is the image of a group homomorphism sending $g$ to $g^p$ for a prime $p$
Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable
Does $a^n \mid b^n$ imply $a\mid b$?
Calculating an improper integral as a limit of a sum.
Irreducibles are prime in a UFD
Intuitive understanding of the Reidemeister-Schreier Theorem
A Schwartz function problem
What's a BETTER way to see the Gauss's composition law for binary quadratic forms?
A subset $P$(may be void also) is selected at random from set $A$ and the set $A$ is then reconstructed by replacing the elements of $P$.
which odd integers $n$ divides $3^{n}+1$?
Finding the basis of a null space

Let $f$ be differentiable and let $\lim_{x \to \infty}f'(x)=+\infty$ prove that: 1) $\lim_{x \to \infty}(f(x)-f(x-1))=+\infty$ and 2) $\lim_{x \to \infty}f(x)=+\infty$.

1) I’ll prove by contradiction. Let $\lim_{x \to \infty}(f(x)-f(x-1))=a$, where $a$ is from and $\Bbb R$. $\lim_{x \to \infty}(f(x)-f(x-1))=\lim_{x \to \infty}\frac{f(x)-f(x-1)}{x-(x-1)}=\lim_{x \to \infty}f'(c)=a$, where c is from $(x-1,x)$ (mean value theorem). As $x$ goes to infinity so does $c$ i.e. we have that $\lim_{x \to \infty}f'(c)=\lim_{c \to \infty}f'(c)=a$. Which is a contradictition since we have that $\lim_{x \to \infty}f'(x)=+\infty$ .

- Equivalence of Rolle's theorem, the mean value theorem, and the least upper bound property?
- if $f$ is differentiable at a point $x$, is $f$ also necessary lipshitz-continuous at $x$?
- Importance of Least Upper Bound Property of $\mathbb{R}$
- The graph of a Borel measurable function $f \colon \mathbb R^{n} \to \mathbb R^{m}$ is a Borel set in $\mathbb{R}^{n+m}$
- Got stuck in proving monotonic increasing of a recurrence sequence
- What do physicists mean with this bra-ket notation?

2) I will also prove this by contradiction. We have that $\lim_{x \to \infty}f(x)=a$ This is equivalent to $\lim_{x \to \infty}f(x)-f(x-1)=0$. I use mean value theorem and get $\lim_{x \to \infty}(f(x)-f(x-1))=\lim_{x \to \infty}\frac{f(x)-f(x-1)}{x-(x-1)}=\lim_{x \to \infty}f'(c)=0$, where c is from $(x-1,x)$. As $x$ goes to infinity so does $c$ i.e. we have that $\lim_{x \to \infty}f'(c)=\lim_{c \to \infty}f'(c)=0$. Which is a contractition since we have that $\lim_{x \to \infty}f'(x)=+\infty$ .

- Infinite Series $\sum\limits_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$
- $L^2$ norm inequality
- Chain rule for second derivative
- Proving that $ f: \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.
- How can we find and categorize the subgroups of $\mathbb{R}$?
- Do all polynomials with order $> 1$ go to $\pm$ infinity?
- Proof that the limit of $\frac{1}{x}$ as $x$ approaches $0$ does not exist
- Behavior of derivative near the zero of a function.

There is no need to prove the statement by contradiction (also notice the issues that the other users pointed out). The fact that:

$$\lim_{x\to +\infty}f'(x) = +\infty$$

implies that for any $x> x_0$, $f'(x)\geq N.$ Since by the Lagrange theorem:

$$ f(x)-f(x-1) = f'(\xi),\qquad \xi\in[x-1,x] $$

for any $x>x_0 + 1$ we have:

$$ f(x)-f(x-1) \geq N, $$

but since $N$ is an arbitrarily big number,

$$\lim_{x\to +\infty} f(x)-f(x-1) = +\infty.$$

The Lagrange theorem also implies:

$$ f(x_0+y)-f(x_0) = y\cdot f'(\xi) \geq Ny, \qquad \xi\in[x_0,x_0+y], $$

hence

$$ f(x_0+y) \geq f(x_0) + Ny $$

gives:

$$ \lim_{x\to +\infty} f(x) = +\infty.$$

1) is wrong because the converse of having an infinite limit is not just the existence of a finite limit. You could also have no limit at all.

2) Same problem.

- Explicit example of Koszul complex
- Definite Integral $\int_2^4\frac{\sqrt{\log(9-x)}}{\sqrt{\log(9-x)}+\sqrt{\log(3+x)}}dx$
- What are some examples of principal, proper ideals that have height at least $2$?
- Validity of conditional statement when the premise is false.
- Proofs of Hyperbolic Functions
- Why are they called “Isothermal” Coordinates?
- Finite expectation of renewal process
- integrate $dx/(a^2 \cos^2x+b^2 \sin^2x)^2$
- Expected value of rolling a $5$ followed by a $6$ different than rolling two consecutive sixes?
- A wedge sum of circles without the gluing point is not path connected
- Proof verification for proving $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ by induction
- Improper integral: $\int_1^{+\infty}\frac{\mathrm dx}{x(x+1)(x+2)\cdots(x+n)}$
- Simple L'Hopital Question
- Find All $x$ values where $f(x)$ is Perfect Square
- Closure of the interior of another closure