# Tagged Questions

Questions on the evaluation of limits.

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### How to find the limit of an algebraic function

The question is to find this limit: $$\lim_{x\to\infty}\frac{2x^\frac{5}{3}- x ^\frac{1}{3}+7}{x^\frac{8}{5} +3x + \sqrt{x}}$$ I need any hint to help since I tried so much and couldn't solve it.
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### How prove this limit $\lim_{n\to\infty}\alpha_{n}=0$

Question: let $$\alpha_{n}(w)=\inf{\{|w\cdot v|:v\in Z^d \mbox{such that} ,0<|v|\le 2^n\}}$$ show that $$\lim_{n\to\infty}\alpha_{n}=0$$ where $d>2$ My try:sinc $$0<|v|\le 2^n$$ then ...
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### How to find this limit:

The question is to find this limit without using H.R (Hopital Rule): $$\lim_{x\to 1}\frac{x^4 -1}{x^3 -1}$$ So this will be $\frac{0}{0}$ Which is indetermined but using H.R we find it is ...
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### 'ϵ-δ' proof for the following sequence

I need help writing a formal 'ϵ-δ' proof for the following sequence: $$\lim_{n\to \infty}(n+2)^2 \sin(1/n)=\infty$$ Thanks in advance.
### Suppose $a_n$ converges and $b_n$ divergent. What can you tell about $a_n+b_n$? [closed]
Suppose $a_n$ converges and $b_n$ divergent. What can you tell about $(a_n+b_n)$?
How to evaluate this limit without using hopital rule: $$\lim_{c\rightarrow + \infty}{\frac{\text{sinh}\sqrt{c}}{2\sqrt{x}}}$$ Here is what I have done so far: we know that $\text{sinh}(x)= ... 1answer 44 views ### Limit of a sum of roots proof Given the sequence: $$a_n=\alpha\sqrt{n+a}+\beta\sqrt{n+b}\ with\ \ \alpha,\beta,a,b\in\mathbb{R}\ and\ \alpha,\beta\neq0$$ Prove that $$\lim_{ n\to \infty} a_n = 0\ iff\ \alpha=-\beta$$ I start ... 4answers 59 views ### Showing that$\lim\limits_{x \rightarrow 0} \frac{1}{x}$does not exist. Forgive me if this has been asked before, but I searched and could not find an answer. I am trying to show that$\lim\limits_{x \rightarrow 0} \frac{1}{x}$does not exist. If the limit did exist, and ... 1answer 39 views ### A short question about an e identity Why is this$\{(1+\frac1{a_n})^{a_n}=e\}$true when:$a_n \to -\inftya_n$is a sequence. Thanks. 1answer 37 views ### 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... 1answer 39 views ### Review of solution: Prove$\liminf({a_n}) \ge \liminf({b_n}){a_n} \ge {b_n}\forall n \in $Prove:$\liminf({a_n}) \ge \liminf({b_n})$I proved it by contradiction. Let's assume$\liminf({a_n}) < \liminf({b_n})$.$a := \liminf({a_n})b := \liminf({b_n})$... 2answers 46 views ### limit problem (with roots) Is it possible to evaluate this limit without graphing or guessing (ie to replace it by a simpler function) $$\lim_{x\to 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}$$ I tried normalizing by multiplying by ... 2answers 55 views ### A Double Limit Question Maybe it's easy, but: Is it true that $$\lim_{(x,y) \rightarrow (0,0)} \frac{x}{\sqrt{x^2+y^2}}=0$$ If it is, could you help me prove it? Thanks 2answers 86 views ### Find$L=\lim \limits_{n\to \infty}\sqrt[n]{x^n+y^n+z^n}$Find the limit following: $$L=\lim \limits_{n\to \infty}\sqrt[n]{x^n+y^n+z^n}$$ With$x,\: y\: z\in R$P.S I think this limit result is$L=max\left\{x,\: y\: z \right\}$. But i'm not find it, so ... 2answers 73 views ### Is the use of L'Hôpital's rule in this limit wrong? I want to find this limit: $$\lim_{h \to 0}\frac{(x+h)^3-x^3}{h}$$ I know that I can expand$(x+h)^3$and with a little of algebra, the limit is$3x^2$. My question: Can I use L'Hôpital's rule to ... 2answers 60 views ### Limit of$f(x)=x^2(1+2+\cdots+\left\lfloor \dfrac{1}{|x|} \right\rfloor) $Suppose$f$a function defined on$\left[\dfrac{-1}{2};\dfrac{1}{2}\right]$as$f(x)=x^2(1+2+\cdots+\left\lfloor \dfrac{1}{|x|} \right\rfloor) $. How can I prove that$f$has a finite limite on$0$? ... 1answer 47 views ### Limit Proof Check Reviewing limits and I'm afraid i may be making mistakes, just looking for a quick proof check.$f(x)=x^4$, prove that$\lim _{x \rightarrow a}=a^4$by showing how to find$\delta$. This is my ... 1answer 39 views ### Prove using Epsilon-Delta and Intermediate Value Theorm Suppose that$f,g\colon[a,b]\to\mathbb R$are continuous functions with$f(a)<g(a)$and$f(b)>g(b)$. Prove that there exists an$x\in(a,b)$with$f(x)=g(x)$. I believe this uses the Intermediate ... 3answers 67 views ### Calculating limit of function To find limit of$\lim_{x\to 0}\frac {\cos(\sin x) - \cos x}{x^4} $. I differentiated it using L Hospital's rule. I got $$\frac{-\sin(\sin x)\cos x + \sin x}{4x^3}\text{.}$$ I divided and multiplied ... 2answers 64 views ### Trouble with l'Hôpital's rule This is an assignment and I am stuck: Find the limit, whether finite or infinite, or indicate that the limit does not exist. Use l'Hôpital's Rule if appropriate. $$\lim_{x\rightarrow 0} ... 0answers 84 views ### Find the limit L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}} Find the limit following:$$L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$$P.S I tried find this limit, but it's made me confused. 0answers 38 views ### Limit on both inside and the range of an Integral$$\lim_{n\rightarrow \infty }\int_{0}^{kn}e^{tx}\frac{1}{k}\left ( 1- \frac{x}{kn}\right )^{n-1} \, \mathrm{d}x$$Can I first show$$\frac{1}{k}\left ( 1- \frac{x}{kn}\right )^{n-1}$$converges on ... 2answers 82 views ### The maximum number of Biological Species? [closed] A simple question that has led to some heated discussion over at Biology Stack Exchange: What is the maximum possible number of Biological Species? or Alternatively: What is the upper limit ... 1answer 34 views ### \lim_{\varepsilon \to 0}\int_{|x|>\varepsilon}\frac{e^{-(t-x)^2}}{x}dx=? I'm computing this integral$$\lim_{\varepsilon \to 0}\int_{|x|>\varepsilon}\frac{e^{-(t-x)^2}}{x}dx=?$$I'm not sure that its integral whether exist. How could I solve it? Thanks for ... 0answers 32 views ### limit of a recursive sequence, Am I allowed to divide by b_n^2?$$b_1 > 0{b_{n + 1}} = {{{b_n}^2 + 1} \over {{b_n}}} = {{{{{b_n}^2} \over {{b_n}^2}} + {1 \over {{b_n}^2}}} \over {{{{b_n}} \over {{b_n}^2}}}}\mathop = {{1 + {1 \over {{b_n}^2}}} \over {{1 ... 1answer 34 views ### Take limit of an Integral (both the limit and the function inside) $$\mathop {\lim }\limits_{n \to \infty } \int\limits_a^{bn} {{f_n}(x)g(x)dx}$$ I am stuck now, but I can show f_n is bounded in the region (a,bn) I tried to to this but I have no idea if this is ... 1answer 38 views ### epsilon delta approach to a problem The following is what I would like to show. $$\lim_{x \to5}\frac{1}{x-3}=1/2$$ Given any$\epsilon>0$,$|\frac{1}{x-3}-\frac{1}{2}|\le\frac{|x-5|}{|x-3|}\le2\epsilon$How do I get rid of ... 2answers 38 views ### Limit at infinity involving$e\$
I am to find the limit of $$\lim_{x \to \infty} \left(1+\frac{x}{5x^3+x^2+8}\right)^ {\dfrac{x^3+8}{x}}$$ I could not find the proper substitution here. I would be happy if someone could shed some ...