Prove by using the formal definition of limit that:   $\displaystyle\lim_{x\to \infty  }\dfrac{1}{x^4 + x^2 + 5} = 0$  Can anyone please lend a hand here?
120410
How would I go about proving this limit?  $\lim_{x\to\infty}\frac{1}{x^4+x^2+5}=0$  so far I have:  $|f(x) -L|&lt; ϵ$ wherever $x &gt; N$  $|\frac{1}{x^4+x^2+5} - 0| &lt; ϵ $ wherever $x &gt; N$  $|\frac{1}{x^4+x^2+5}| &lt; ϵ  ,\; x &gt; ∞$, assuming, $x &gt; 0$  taking the absolute value $\frac{1}{x^4+x^2+5} &lt; ϵ$  ${x^4+x^2+5}$ > $\frac{1}{ϵ}$  i am not sure whether this is correct, but what would be the next step?
29010
Given $\lim \limits_{x\to\infty}(1+\frac{1}{x})^{x}$, why can't you reduce it to $\lim \limits_{x\to\infty}(1+0)^{x}$, making the result "$1$"?  Obviously, it's wrong, as the true value is $e$.  Is it because the $\frac{1}{x}$ is still something even though it's really small?  Then why is $$\lim_{x\to\infty}\left(\frac{1}{x}\right) = 0\text{?}$$  What is the proper way of calculating the limit in this case?
38578
Prove that:  $$\lim_{n\rightarrow \infty}\dfrac {n^{2}}{2^{n}}=0$$  I think I need to show $2^n \geq n^3$ $\forall n \geq10$. Is it true?  What about: $$\lim_{n\rightarrow \infty}\dfrac {n^{2}}{n!}=0$$  What to do with $n!$?
62679
$$\lim_{x\to4} \frac{1}{x-4}$$ Would it be correct to say that the limit is undefined because the denominator would be $0$?
29585
$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$  It's clear that we can simplify the limit a little bit, after which we get:  $\lim\limits_{n\to\infty}{(\frac{1}{4})^{n}\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{4}{5})^{k}}$  I could further simplify the expression, but I feel like there's a more elegant solution.   Give me a hint, please
117927
$$\lim_{x\to \infty}  {\frac{x}{x^2+1} +\frac{x}{x^2+2} + ... + \frac{x}{x^2+x} }$$  It seems obvious that the result is zero for each term but in order to break the limit into its individual  parts we must know that every term's limit exists .
87774
How do I prove that:   $\lim \limits_{n\to \infty}2^{1/n}=1$  Thank you very much.
96491
I have a strong feeling that the following limit is zero, can anybody help me prove it.    $ \lim\limits_{(x,y,z) \to (0,0,0)} \frac{{xyz}}{{x+y+z}}$   Thanks!
48503
How to formally prove that if $\lim \limits_{n \to \infty}a_n=\infty$, then  $\lim \limits_{n \to \infty}\frac{1}{a_n}=0$.  I got confused for how to mix between convergence to infinity and having finite limit. how do I write a proof to link between two things? ( by using $\epsilon $) and not just giving an example.  my proof:   let $M= \frac{1}{\epsilon}$. there exists such a $N\in\mathbb N$ s.t for every $n&gt;N$, it is true that: $$a_n&gt;\frac{1}{\epsilon}$$  we should prove that for every $\epsilon&gt;0$ there exists such a  $N\in\mathbb N$, s.t for every $n&gt;N$ $$\left|\frac1{a_n}\right|&lt;\epsilon$$  so we take the N that satisfies the first conclusion, and that will mean for every $n&gt;N$$$\left|\frac1{a_n}\right|&lt;\epsilon$$
99850
$$\lim_{x\to 0}\frac{10^x - 2^x - 5 ^ x + 1 } {x\tan x} $$ This is an indeterminate limit.  I want help in solving this problem. Thanks in advance
78547
Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?:  $\forall$ $\delta &gt;0$ $\exists$ $\epsilon&gt;0$ such that if $0&lt;|x-a|&lt;\delta$ then $|f(x)-L|&lt;\epsilon$  I've been trying to solve this for a while, and I think it would give me a greater understanding of why the limit definition is what it is, because this alternative definition seems quite logical and similar to the real one, yet it supposedly shouldn't work.
97971
$$\lim_ {n\to \infty}\ \frac{n}{n+1}=1$$   I'd write it like this:  $$\lim_ {n\to \infty}\ \frac{n}{n}+\frac{n}{1}=\lim_ {n\to \infty}\ 1+n=\infty$$  What am I missing?
48461
How to prove the following limit?   $$\lim_{n \to \infty} (1+1/n)^n = e$$    I can only observe that the limit should be a very large number!   Thanks.
108485
How can I prove that $\displaystyle{\lim_{(x, y) \to (0, 0)} \frac{x^3 - y^3}{x^2 + y^2}}=0$?  The method I've been taught is the pinching one, where you compare the absolute value of the function to greater limits that are known to equal zero, but I haven't managed to find a comparison that works.  Could someone point me in the right direction?
87671
How to use the formal definition of the limit of a sequence to prove the following limit?  $\lim_{x \to -2} {(x^3 + 4x^2+ 4x-1)} = -1,$  $\lim_{x\to 1}{(\frac{1}{x})} = 1.$
31983
If $-1&lt;x&lt;1$ show that $\lim\limits_{n \to \infty}{n\,x^n} = 0$. I don't have idea. I only prove that $n\,x^n$ is decreasing.
103696
Could someone give me a hint? I would like to continue to attempt it. The limit to evaluate that I would like a hint on is:  $$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$$