Tagged Questions
Questions on the evaluation of limits.
0
votes
0answers
4 views
Find where the limit does not exist for the function
Given the function: $f(x,y) = \frac{xy^4}{x^2+y^8}$, find a path where the limit does not exist at the origin.
I am having problems with this because of lot of paths go to $0$ but I know the limit ...
1
vote
0answers
26 views
Generating function solution to previous question $a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor}$
In attempting to answer
this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying ...
0
votes
2answers
21 views
Is this function continuous at the origin
I have just started learning to use the two-path test to find limits and I am very doubtful of my ability so I am verifying what I have done below is correct.
I first tried letting y=0 and x -> 0 ...
0
votes
3answers
23 views
Showing limit does not exist using two-path test
I am new to using two-path test and my textbook only discusses it without showing any examples. I attempted to do this question below but I am not sure if I am correct. The question says to show the ...
0
votes
0answers
21 views
Large $r$ limit of a certain integral
I'm interested in computing the following integral
$$\lim_{r \to \infty} \int_{-1}^1 dt f(t) e^{i r (t-1)} $$
I know nothing about the form of the function $f(t)$. I simply want to make a statement ...
8
votes
1answer
61 views
Find asymptotic of recurrence sequence
Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$.
The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$.
...
1
vote
1answer
27 views
The limit of the product of functions that tend to infinity and zero
If $x$ is fixed within $[0,1]$, the limit of $n^3x^n(1-x)$ as n tends to infinity is $0$.
But how do I show this? $n^3$ tends to inifinity, but the other term tends to zero, and surely I cannot ...
1
vote
2answers
30 views
Closed form of the limit of a sequence (weighted average)
I have a sequence, which can actually be seen as Riemann-Stieltjes integration with a binomial distribution. $\rho \in (0,1)$.
$$
S_N ...
0
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0answers
11 views
Evaluating limit involving hypergeometric function $\lim_{z\rightarrow1^-}{}_2F_1(-a,-b,-(a+b),z)$
The following expression involving a hypergeometric function popped out as a solution to an integral that I've been working on (via Gradshteyn and Ryzhik):
...
2
votes
2answers
31 views
Making a multivariable Function continuous
This function
$$f(x,y)=\frac{e^{xy}-\cos (x)+\sin(xy)}{x}$$
can be made continous for $f(0,y)$ by defining
$$f(0, y) = 2y .$$
My question is: how can i get to this conclusion ("$2y$ must be it") ...
2
votes
5answers
66 views
Prove that a function does not have a limit as $x\rightarrow \infty $
Problem statement:
Prove that the function $f(x)=\sin x$ does not have a limit as $x\rightarrow \infty $.
Progress:
I want to construct a $\varepsilon -\delta $-proof of this so first begin by ...
1
vote
2answers
32 views
Integrand becomes infinite when length of interval goes to zero
So I want to evaluate the following integral
$$ \int_{- \pi }^{\pi } | \theta |^{ \alpha} e^{x ( \cos \theta -1) } d \theta$$
when $\alpha < 0$, and look at the limit as $ x \rightarrow +\infty ...
2
votes
2answers
33 views
upper bound of sequence of functions
What function will be an upper bound of the sequence
$f_{n}(x)=\Bigl(1-\frac{x}{n}\Bigr)^{n}\ln(x)$
since $\displaystyle\lim_{n\to\infty}\Bigl(1-\frac{x}{n}\Bigr)^{n}=e^{-x}$ so we should probably ...
1
vote
0answers
27 views
Limits along what curves suffice to guarantee the existence of a limit?
For functions f from $R^2$ to $R$, we can define the limit of $f ( x,y)$ as $(x,y)$ goes to $(a,b)$ along the curve $C$ for any continuous curve $C$ passing through (a,b). And it is a theorem that if ...
1
vote
4answers
56 views
Find the limit of a recursive square root sequence.
Find the limit of the sequence $$\left\{\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \dots\right\}$$
Another way to write this sequence is $$\left\{2^{\frac{1}{2}},\hspace{5 pt} ...
3
votes
1answer
53 views
Math Differentiation Limits
$$\lim_{x\to 3} \frac{2x^2 + 7x-15}{x-3}$$
What I Simplified
Step 1 : $\frac{2x^2 + 10x -3x -15}{x-3}$
Step 2 : $\frac{2x(x + 5)-3(x + 5)}{x-3}$
Step 3 : $\frac{(2x - 3)(x + 5)}{x-3}$
but unable ...
2
votes
1answer
42 views
Use the ε-δ definition of a limit to prove this.
I know to how prove normal limits using the epsilon-delta definition, say:
limx→a f(x)=L
But, there was a question on my textbook which I couldn't quite figure out to do, even though I've thought ...
0
votes
2answers
60 views
How to compute $\lim_{n\to\infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}3\right)^n$?
How would one compute $$\lim_{n\to\infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}3\right)^n$$ if $a,b,c>0$.
I've never done an integral 3 variables before! This is from a chapter called Interchange of ...
0
votes
3answers
36 views
When computing $\lim_{x\to -\infty}\frac{2x+7}{\sqrt{x^2+2x-1}}$, I don't get -2.
This limit:
$\lim_{x\to -\infty}\frac{2x+7}{\sqrt{x^2+2x-1}}$
is supposed to be equal to -2. My textbook and Wolfram Alpha both state that. However, I can't seem to get same exact result.
...
1
vote
1answer
39 views
Does $\lim \limits_{(x,y)\to(0,0)} \frac{(x+y)(y + (x+y)^2)}{y - (x+y)^2}$ exist?
Evaluate
$$\lim \limits_{(x,y)\to(0,0)} \frac{(x+y)(y + (x+y)^2)}{y - (x+y)^2}$$
I can see that the repeated limits are both zero but I'm unable to prove that the simultaneous limit exists and is ...
0
votes
1answer
48 views
Use the $\varepsilon$-$\delta$ definition of a limit to prove this.
I know to how prove normal limits using the epsilon-delta definition, say:
$$\lim_{x\to a}f(x) = L$$
But, there was a question on my textbook which I couldn't quite figure out to do, even though ...
0
votes
1answer
47 views
Prove: If $\lim_{n\rightarrow\infty}|a_n| = 0$, then $\lim_{n\rightarrow\infty}a_n = 0$
Using the squeeze theorem, prove the following:
If $\lim_{n\rightarrow\infty}|a_n| = 0$, then $\lim_{n\rightarrow\infty}a_n = 0$.
Let $f(x) = |a_n|$. Let $g(x) = a_n$ such that $\forall x : ...
2
votes
2answers
48 views
Computing the limit
I usually use L'Hopital's rule to compute a limit, but in these two cases, I don't know how to break it down.
a) $\lim \limits_{n\to\infty}(2^n+3^n)^{1/n}$
b) $\lim ...
0
votes
2answers
18 views
How does variable substiution in limit finding work?
I am asked to calculate the following limit:
$\displaystyle \lim{x \to \infty}$ for $x \ln(1+\frac{1}{x})$. This is an indeterminate form of $\infty * 0$. I have made a lengthy calculation switching ...
0
votes
2answers
27 views
Simplying radicals for limit leads to extra minus symbol
I am asked to calculate the limits of $\dfrac{x}{\sqrt{1+x^2}}$ for $x\to\infty$ and $x\to-\infty$
When I simplify the radical for $x\to\infty$, I generate:
$\dfrac{x}{\sqrt{1+x^2}} = ...
3
votes
2answers
34 views
Prove $\sqrt{s_n+1} = \frac{1}{2}(1+\sqrt{5})$
This is to prove how the limit of $s_n$ converges to $\frac{1}{2}(1+\sqrt{5})$.
Assume: $s_1 = 1$; for $n \geq 1$, $s_{n+1} = \sqrt{s_n + 1}$.
How to prove this converges to ...
1
vote
0answers
23 views
A problem with a limit of two variables
I'm trying to find out this limit, if it exists. If yes, what is its value? If not, how do I show?
$$\lim_{(x,y)\to(0,0)} \frac{\sin^{⁻1}(y)+\cos^{-1}(x)}{xy}$$
I tried the few ways I know, like ...
1
vote
3answers
37 views
Use the limit of the derivative to find the slope of the tangent line to the graph of $y=1/x^2$ at the point $(1,1)$
I'm having major difficulty understanding derivatives and limits. I need help.
0
votes
1answer
19 views
one sided derivatives
Show that if $ f'(a^+) $ and $f'_+(a) $ exist, then $ f'(a+) = f'_+(a) $.
Here $ f'(a+) = \lim_{x \to a^+} f'(x) $ and $ f'_+(a) = \lim_{x \to a^+ } \frac{f(x) - f(a)}{x-a} $
$\textbf{Attempt:}$ ...
0
votes
5answers
97 views
Find $\lim_{x \to 0^+} \frac{\ln{3x}}{\ln{2x}}$ without l'Hospital's rule
How do you find the following limit without using l'Hospital's rule???
$$
\lim_{x \to 0^+} \frac{\ln{3x}}{\ln{2x}}
$$
0
votes
2answers
29 views
finding function with limit and derivative
The limit is:
$\lim_{x\to3}\frac{7^{3x}-7^9}{x-3}$
This is equal to the derivative of a function f at a point a (that is, f′(a)).
how do you figure out what the function f is and the value of a ?
I ...
3
votes
3answers
59 views
Calculus limit homework problem
$$ \lim_{n→\infty} \frac1 n \left(\left(a + \frac 1 n\right)^2 + \left(a + \frac 2 n\right)^2 + ... + \left(a + \frac{n-1}{n}\right)^2\right)$$
$$ \text{hint: }\ 1^2 + 2^2 + ... + n^2 = ...
0
votes
0answers
15 views
Interchange of Limit Operations Advanced Calc 2 Question
Let $c_0, c_1, c_2,\ldots \in \mathbb{R}$.
Prove that the radius of convergence of the power series $$\sum_{n=0}^{\infty}c_nx^n$$ is $$\frac{1}{\lim_{n\to\infty}\sup\sqrt[n]{|c_n|}}.$$
Also show ...
0
votes
4answers
36 views
What is $\lim_{x\rightarrow0}(1-\tan2x)^{\cot x}$?
$$\displaystyle \lim_{x\rightarrow0}(1-\tan2x)^{\cot x}$$
I see that it has something to do with exp but nothing more.
1
vote
1answer
20 views
Advanced Calc 2 Interchange of Limit Operations Question
Let $a_1, a_2, a_3,...$ be a sequence of non-negative real numbers, let $S_1, S_2, S_3,...$ be a sequence (finite or infinite) of disjoint nonempty sets of natural numbers whose union is ...
1
vote
2answers
31 views
Problems with the existence of limits of several variables
I started some time studying calculus of two variables, and I'm having difficulty in knowing (and prove) that a limit does not exist, how could I resolve, for example, these, whose statement asks to ...
2
votes
2answers
42 views
Indeterminate form from calculus
What do we mean by Indeterminate form ?
can we show $0/0$ anything as we wish
i mean is it not unique
there can be several answers for it
0
votes
1answer
43 views
How can I prove that this limit is $+\infty$? [on hold]
How can I prove that this limit is $+\infty$ ?
$$
\lim _{x\rightarrow0} \frac{\cos(x)}{ {|x|}} \>.
$$
0
votes
1answer
44 views
0
votes
0answers
19 views
The limit of a composite function
I am presented with the following task:
"You are given that $k(h) \neq 0$ and $h \neq 0$. If $\lim_{k \to 0} F(k) = L$ and $\lim_{h \to 0} k(h) = 0$, show that $\lim_{h \to 0}F(k(h)) = L$.
This is ...
2
votes
3answers
42 views
How does the chain rule for limits work?
I have to evaluate the limit of this function,
$$\lim_{x\to0^+} \arctan(\ln x)$$
I already know the answer, it's $-\dfrac{π}{2}$, but the only part I don't get it, how does it come to that? I did ...
2
votes
3answers
46 views
Evaluate the Limits that Exist without L'Hopital's Rule
I have four limit problems, for homework, but I don't quite understand them. I will only ask for help on one of them because that should be all I need to understand these. According to the Calculus ...
2
votes
3answers
67 views
How to prove that $\lim_{n\rightarrow\infty} (1-\frac{k}{n})^n = e^{-k}$ and $\lim_{n\rightarrow\infty} (1-\frac{k}{n})^k = 1$?
How to prove that
$\lim_{n\rightarrow\infty} (1-\frac{k}{n})^n = e^{-k}$
and
$\lim_{n\rightarrow\infty} (1-\frac{k}{n})^k = 1$?
Any answer will be appreciated. Thanks.
6
votes
3answers
86 views
Calculating the limit of $\sum_{k=1}^{n}\frac{1}{4^k\cos^2\frac{x}{2^k}}$
How i can prove that $$\lim_{n \to\infty}\sum_{k=1}^{n}\frac{1}{4^k\cos^2\frac{x}{2^k}}=\frac{1}{\sin^2(x)}-\frac{1}{x^2}$$.
0
votes
1answer
15 views
A limit involving the Hurwitz/generalized Riemann zeta function
I want to show that
$$ \lim_{s \to 1} \Big( \zeta(s,a) - \frac{1}{s-1} \Big) = - \psi(a)$$
where $\zeta(s,a)$ is the Hurwitz zeta function and $\psi(a)$ is the digamma function.
The only thing I ...
0
votes
1answer
28 views
How find this limit $I=\lim_{x\to 0}\left(\frac{a^x-x\ln{a}}{b^x-x\ln{b}}\right)^{\cot^2{x}}$
Find the limit of the value
$$I=\lim_{x\to 0}\left(\dfrac{a^x-x\ln{a}}{b^x-x\ln{b}}\right)^{\cot^2{x}},(0<a\neq1,0<b\neq 1,a\neq b)$$
Following is My ugly methods
...
4
votes
1answer
34 views
Can the exponential function be reprsented as infinite product?
Is there any representation of the exponentil function as infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e.
$$\mathrm ...
-2
votes
2answers
50 views
A question on Limits [on hold]
How can you find this limit? (or show that a limit doesn't exist)
$$ \lim_{x \to 0} \dfrac{(x^2)(\cos(1/x))}{(2x+4)}$$
-1
votes
0answers
28 views
Need help with limits. [on hold]
Find the limit or show that it doesn't exist: lim x approaches a (x^2+3x)signx/2x+4 for a=1, a=-2, and a=infinity
-1
votes
2answers
52 views
Applying L'Hopital's rule to $\lim\limits_{x \to 0}\frac{2}{x^2}$
$$\lim_{x \to 0} \frac{2}{x^2}.$$
If we apply L'Hopital rule, Then the procedure would go: $0/2x$, and then $0/2$ which is zero. What is wrong with this application L'Hopital rule, as it clearly ...
