Intereting Posts

Proving the multiplicativity of a binary quadratic form
How many triangles in picture
(Easy?) consequence of the Riemann Hypothesis
Learning mathematics as if an absolute beginner?
1/1000 chance of a reaction. If you do the action 1000 times, whats the new chance the reaction occurs?
The cycle space and cut space are orthogonal complement
Multivariate differentiability verification
Find a closed form for this infinite sum: $ 1+\frac 1 2 +\frac{1 \times2}{2 \times 5}+\frac{1 \times2\times 3}{2 \times5\times 8}+ \dots$
Solving the Diophantine Equation $2 \cdot 5^n = 3^{2m} + 1$ over $\mathbb{Z}^+$.
Probability distribution of the maximum of random variables
Every prime ideal is either zero or maximal in a PID.
How to find a normal abelian subgroup in a solvable group?
Linear w.r.t. any measure
General Continued Fractions and Irrationality
Gradient Estimate – Question about Inequality vs. Equality sign in one part

Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$

I’m trying to solve it by using the squeeze theorem but I’m stuck.

I’m looking for a function $g(x)$ such that $g(x) \leq x\lfloor\frac{1}{x}\rfloor \leq x(\frac{1}{x}) = 1$

- Integration of exponential and square root function
- Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$
- limit of a sequence. might be related to Cesaro theorem
- Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?
- Prove an inequality
- how to compute the de Rham cohomology of the punctured plane just by Calculus?

Can anybody give me a hint?

- Integrate $e^{-\frac{y^2}{2}}\left(\frac{1}{y^2}+1\right)$
- Methods to solve differential equations
- The difference between pointwise convergence and uniform convergence of functional sequences
- Another Epsilon-N Limit Proof Question
- Why is the second derivative of an inflection point zero?
- Prove that $f(x)=0$ has no repeated roots
- A continuous function on $$ not of bounded variation
- How does one evaluate $\lim\limits _{n\to \infty }\left(\prod_{x=2}^{n}\frac{x^3-1}{x^3+1}\right)$?
- Sine and cosine series
- Proving a limit of a trigonometric function: $\lim_{x \to 2/\pi}\lfloor \sin \frac{1}{x} \rfloor=0$

By definition,

$$\frac{1}{x} – 1 < \left\lfloor \frac{1}{x} \right\rfloor \leq \frac{1}{x}.$$

- If $x > 0$, this implies $1 – x \le x \lfloor \frac{1}{x} \rfloor \le 1$, so by squeezing you get $\lim_{x \to 0^+} x \lfloor 1/x \rfloor = 1$.
- If $x < 0$, the inequalities are reversed: $1 – x \ge x \lfloor \frac{1}{x} \rfloor \ge 1$ and again by squeezing it follows that $\lim_{x \to 0^-} x \lfloor 1/x \rfloor = 1$.

The RHS limit and the LHS limit exist and are equal, thus $\lim_{x \to 0} x \lfloor 1/x \rfloor = 1$.

Here is an approach. We recall the identity

$$ \lfloor y\rfloor = y – \{y\} $$

where $ \{y\} $ is the fractional part of $y$ which has the property $ \{y\}<1. $

Let $y=\frac{1}{x}$ so the limit becomes

$$ \lim_{y\to \infty} \frac{y – \{y\}}{y} = 1-\lim_{y\to \infty} \frac{\{y\}}{y}=1-0=1. $$

- Calculating probability of 'at least one event occurring'
- How to integrate an exponential raise to the inverse sine?
- Polynomial Question
- General Proof for the triangle inequality
- Show that the sequence $\sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}},…$ converges and find its limit.
- Why aren't all important differential equations solved and codified once and for all?
- Derivative of function with 2 variables
- Reference book on measure theory
- basic induction probs
- prove inequlity about cardinality power sets
- If $A^2\succ B^2$, then necessarily $A\succ B$
- How to apply reduction of order to find a 2nd linearly independent solution?
- Question about a proof on Atiyah Macdonald
- Does there exist an unbounded function that is uniformly continuous?
- When is the image of a proper map closed?