# Tagged Questions

39 views

### How to evaluate this limit with l'hopital's rule

is it possible to use L'hopital for this or is there another method I'm missing? I have no idea how to even start this. $$\lim_{x\to \infty} \frac{(9x+1)^\frac12}{x+1}$$
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### Evaluate limit with L^Hopital's rule if possible

I'm not sure how to use the L'Hopital's rule properly. The question is $$lim_{x\to \infty} \frac{x^2 + \sin x }{x^2}$$ I tried replacing x by $\frac{1}{t}$ and limit to $lim_{t\to 0}$ and use ...
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### If $f(x)=\chi_{(0,\infty)}\exp(-1/x)$, show that $f\in C^{\infty}$.

Define the function $f:\mathbb{R}\to\mathbb{R}$ as follow: $f(x)=\chi_{(0,\infty)}\exp(-1/x)$ In other words: $f(x)=0$ if $x\le 0$, and $f(x)=\exp(-1/x)$ if $x>0$. Show that $f\in C^{\infty}$. ...
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### How should I define the limit definition of a derivative using negative numbers?

Typically the derivative is defined at a point $x$, assuming it is differentiable at it, by $$\lim_{n \rightarrow \infty} \frac{f(x + \frac{1}{n}) - f(x)}{\frac{1}{n}}$$ ...
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### Derivative Proof using Basic Defintion

I have to use the basic definition of the derivative to find the derivative of $$f(x)=\frac{1}{\sqrt{x}}$$ for x>0 I need to use the limit... $$\lim_{x \to c}\frac{f(x)-f(c)}{x-c}$$ So I have ...
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### Using definition of derivative to find $\sqrt{11\theta}$

$$\lim_{h \to 0} \frac{f(\theta+h)-f(\theta)}{h} \;\;\;\;\;\;\;\;\;\;\; \textbf{(1)}$$ $$\lim_{z \to \theta} \frac{p(z)-p(\theta)}{z-\theta} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \textbf{(2)}$$ ...
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### Question about $f$ continuous function with these conditions?

Suppose I have a differentiable and bounded function $$f: [0, + \infty) \longrightarrow \mathbb{R}$$ such that $$\forall x \in [0, + \infty) \, : f(x) \cdot f'(x) > \sin x.$$ The question is: ...
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### Complex differentiability equivalent to linear approximation

Let $G \subset \mathbb C$ be an open set and $f: G \to \mathbb C$ a complex function on $G$. Prove that the function $f$ is complex differentiable at a point $z \in G$ if and only if there exists a ...
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### Differentiation, from first principles

I am having problems with this question, it would be wonderful if someone can help. Given that $f(x)= x^2 + x - 3$ 1) Find $f(x + h)$ 2) Then express $f(x+h)-f(x)$ in its simplest form 3) Deduce ...
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### How do you prove $e^{-a}=a$ without using graphs?

We're doing a section on limits, continuity, and differentiation in my Advanced Calculus class, and I am at a loss for how to prove this...
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### question about the limit $\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin(x)}{h}$

Because $\sin'(x)=\cos(x)$ we can prove that $\arcsin'(x)=\frac{1}{\sqrt{1-x^2}}$. but, by definition we have $$\arcsin'(x)=\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin(x)}{h}\tag{1}$$ therefore, ...
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### Can someone example and give an example?

Given an example of a function f such that lim f(x) when x approaches infinitive exists, but lim derivative of f(x) when x approaches infinitive does not exist.