Tagged Questions

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Limit of expression with radicals [closed]

$$\lim\limits_{x \to a} \frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}$$ I know I should reverse the sign of the denominator to get the conjugate. However, I'm not sure whether I should multiply ...
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L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
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Here is the limit I'm struggling with: $$\lim_{x\to0}\cfrac{x\tan x-x\sin x}{x\sin^2x/\cos x}.$$ Worked so hard to find it, but couldn't.
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When can't $dy/dx$ be used as a ratio/fraction?

By searching this question, I found this: Can I ever go wrong if I keep thinking of derivatives as ratios? However, the answers don't have what I'm looking for! (Edit: Meaning, a counterexample. ...
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$\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)}$?

Is it always true that: $\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)}$ where $m,k \in \mathbb N$ ? I tried it with a few numbers and it seems to work every time.
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Integration involving $\log_2(x)$

Having a hard time going about this problem: $$\int{\frac{\ln(2)\log_2(x)}{x}}$$ I believe $\ln(2)$ would be considered a constant, so than the equation would then changed to: ...
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Solve algebraically: $\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$

$$\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$$ The professor says we can't use l'hopital's rule and must solve algebraically.
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Recognizing the proper polynomial factorization to solve an indeterminate limit

I had to solve the $\lim_{x \to 3} \frac{x^3-3x^2-x+3}{x^2-x-6}$ that is indeed an indeterminate form ($\frac{0}{0}$). The approach I adopted was to factor the polinomials so that I can deviate from ...
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I have an expression that I need to simplify, I know the answer (wolframalpha) but I'm not sure of the rule that gets me there. $\dfrac{(\alpha) X_1^{\alpha -1} X_2^{1-\alpha}}{(1-\alpha)X_1^\alpha ... 0answers 21 views I need to find$n$that$\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$So what I did is this:$(n+1)\ln(n+1) > 1000 \Rightarrow n>190$When I put it back I see that$\frac{1}{192 \cdot ln(192)} \not < 10^{-4}$. ... 6answers 285 views Why Not Define$0/0$To Be$0$? For every number$x$,$x\times 0=0$, hence$\dfrac{0}{0}$can be any number! So$\dfrac{0}{0}$"is knows as indeterminate" [1]. But what if we define it to be$0? I already have an answer, but ... 0answers 21 views Calculating summary with variable multiplication factor I have a formula of thermal conductance heat transfer rate. Here it is: $$Q = \lambda{S (T_1 - T_2) \over L} \Delta t$$ For my calculations I have got some constant values available $$Q = 0.58{1 ... 2answers 95 views Calc 101 Question on simplifying a fraction$$\lim_{h \to 0} \left(\frac 1h -\dfrac{1}{h^2+h} \right).What do I do about the denominators? 1answer 336 views Ceiling to Floor Function Conversion Proof I am working on a proof to convert a ceiling of a fraction to a floor of a fraction. I found this: \begin{aligned} q=\left\lceil \frac{n}{m} \right\rceil \;&\Leftrightarrow\; \frac{n}{m} \leq q ... 2answers 59 views Separating \frac{1}{1-x^2} into multiple terms I'm working through an example that contains the following steps:\int\frac{1}{1-x^2}dx=\frac{1}{2}\int\frac{1}{1+x} - \frac{1}{1-x}dx\ldots=\frac{1}{2}\ln{\frac{1+x}{1-x}}$I ... 4answers 411 views Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication? Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication? I had also got a clue: it's related to e. Please help! ... 2answers 968 views How does he get a perfect swap numerator and denominator. I'm going through a exercise, in which all the answers are given, but the tutor makes a step and I can't follow at all. A massive jump with no explanation. Here is the question:$\lim_{x \to 2} ...
I think the following two inequalities are true. However, the proof may not be easy. Does anyone have any hints? Thank you very much! Fix $a>1$. there exists two constants $K_1$ and $K_2$, such ...
So i have to evaluate this sum: $\displaystyle \frac{1-2^{-2}+4^{-2}-5^{-2}+7^{-2}-8^{-2}+10^{-2}-11^{-2}+\cdots}{1+2^{-2}-4^{-2}-5^{-2}+7^{-2}+8^{-2}-10^{-2}-11^{-2}+\cdots}$ it has the form : ...