Homework Statement
Find tangent line of y=xe^{\frac{1}{x}} at point x=\alpha and it's limit position when \alpha \rightarrow +\infty.
Homework Equations
Tangent of y=f(x) at point M(x_0,f(x_0)): yy_0=f^{'}(x_0)(xx_0)
The Attempt at a Solution
Applying the above equation for tangent of...
Homework Statement [/b]
Determine if the limit exists as a number, ∞, ∞ or DNE
lim x>4 (2)/(sqrt(4x))
The Attempt at a Solution
lim x>4....
I honestly don't know how to solve. Because I don't know what to do with the sqrt function. If someone could lead me in the right direction here...
Homework Statement
##f(x)## is a continuous and differentiable function. ##f(x)## takes values of the form ##^+_\sqrt{I}## whenever x=a or b, (where ##I## denotes whole numbers) ; otherwise ##f(x)## takes real values. Also, ##f(a)\le f(b)## and ##f(c)=1.5##. Graph of ##y=f(x)f'(x)##:
The...
I want to evaluate \displaystyle\lim_{(x,y)\to(1,0)}\frac{y^4(x+1)}{x+1^3+2y^3}
With some help, I was able to prove that the limit is 0, using Hölder's inequality. Like this:
\left(x+1^3\right)^{1/5}\left(\frac{1}{2}y^3\right)^{4/5}\leq\frac{1}{5}x+1^3+\frac{4}{5}\frac{1}{2}y^3...
Homework Statement
Given that
lim f(x) = 4 and lim g(x) = 6
(All limits x > +infinity)
Find the limit
lim [f(x) + 2g(x)]
Homework Equations
The Attempt at a Solution
So I substituted the values of f(x) and g(x) in the equation...
< Mentor Note  thread moved to HH from the technical math forums, so no HH Template is shown >
This is from the question list at the UC Davis Website epsilon delta exercise list.
In the exercise list we have:
Prove that
Which concludes with:
Thus, if , it follows that...
I've been searching around trying to understand them. About to take calculus and I want to be prepared. Could someone explain what they are and give a few typical limit problems and solve them
Thank you
I claim that if a function ##f:\mathbb{R}\rightarrow\mathbb{R}## is continuous at a point ##a##, then there exists a ##\delta>0## and ##h<\frac{\delta}{2}## such that ##f## is also continuous in the ##h##neighbourhood of ##a##.
Please advice if my proof as follows is correct.
Continuity at...
Homework Statement
Prove that :
$$\lim_{x\to 0}\frac{tan^{1}x}{x}<1$$
And
$$\lim_{x\to 0}\frac{sin^{1}x}{x}>1$$
Homework Equations
None
The Attempt at a Solution
I have to prove that arctanx has to be lesser than x.
It's derivative is 1 at x=0 and keeps decreasing as x increases. So it's...
Homework Statement
Hello, thank you in advance for all help. This is a limit problem that is giving me a particularly hard time.
Homework Equations
For what values of a and b is f(x) continuous at every x? In other words, how to unify the three parts of a piecewise function so that there are...
Let ## f(x)=\begin{cases}1 &if \ x=\frac{1}{n} \ where \ n\epsilon \mathbb{Z}^{+}\\
0 & \mbox{otherwise}\end{cases}##
(i) Show that ##c\neq 0## then ##\lim_{x \to c}f(x)=0##
(ii) Show that ##\lim_{x \to 0}f(x)## does not exist.
I attempted to answer the question:
I think we have to...
Homework Statement
Let an → 2. Prove from first principles (i.e. give a direct εN proof) that an2 → 4.
Homework Equations
The Attempt at a Solution
I have tried considering an22 and considering that an24 = (an+2)(an2) but I could not get either of these methods to work. Would...
lim(9x) as x>4 = 5
I thought I was supposed to do this:
94=5
5=5
But apparently I was supposed to use delta and epsilon?
I'm not sure how to find either of these. I know you find epsilon first but I'm really confused so if anyone knows just HOW to find it, that would be extremely helpful...
Given a function f(x), a point x0, and a positive number E (epsilon), write the limit then find delta>0 such that for all x 0< xx0 < delta > f(x)L < E
f(x) = 32x, x0=3, E=.02
Here is my attempt:
Lim (32x) as x>3 = 3
.02 < 32x  3 <.02
.02 < 2x < .02
.01 > x > .01
2.99 > x3...
The website rules say that when people help they are not allowed to include answers? But how am I supposed to check my answers... anyone else have this problem?
f(x) = x^3  12x^2 + 44x  46
x from 1 to 7
The attempt at a solution:
f(1) = 13
f(2) = 2
f(4) = 2
f(5) = 1
f(6) = 2
So naturally, the answer should be: (1,2) U (4,5) U (5,6)
right? Well, it didn't accept this answer. I think there is something wrong with whatever that is accepting the...
Homework Statement
I need help finding the limit of the differential equation.
(dx/dt) = k(ax)(bx) that satisfies x(0)=0
assuming
a) 0<a<b and find the limit as t>infinity of X(t)
b) 0<a=b and find the limit as t>infinity of X(t)
Homework Equations
none
The Attempt at a Solution
I...
I was going through some important points give in my textbook and I saw this:
##\log_e x < \sqrt x##
How did they get this?
I know calculus so you can show this using differentiation, etc.
One possible way is that they took
##f(x)=\sqrt x\log_e x##
And tried to prove it is always greater than zero.
Homework Statement
Using sandwich theorem evaluvate:
$$\lim_{x\rightarrow \infty} \frac{x+7sinx}{2x+13}$$
Homework Equations
Sandwich theorem
The Attempt at a Solution
##7 \leqslant 7sinx \leqslant 7##
##x7 \leqslant x+7sinx \leqslant x+7##
Now my doubt: I want to divide the expression by...
Homework Statement
In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(∞,∞)? Is there any specific reason behind this notation i.e. does the variable have...
Homework Statement
[/B]
I have to find the radius of convergence and convergence interval. So for what x's the series converge.
The answer is supposed to be for every real number. So the interval is: (∞, ∞).
So that must mean that the limit L = 0. So the radius of convergence [ which is...
Homework Statement
Evaluate the limit
1 1 1
lim ∫ ∫ ... ∫ cos^2((pi/2n)(x1 + x2 +... xn))dx1 dx2 ... dxn
0 0 0
n→∞
Homework Equations
Well, I know that we can change this using a double angle rule, so that the integrals become 1/2 + 1/2 cos (2*pi/2n)(x1...
Homework Statement
I am studying for a calculus test tomorrow on this website (http://archives.math.utk.edu/visual.calculus/6/index.html). I am working on the limit comparison test problems but I am unfamiliar with the form they use in their solutions. For example:
Limit comparison test (prove...
Mod note: Fixed the LaTeX. The closing itex tag should be /itex, not \itex (in brackets).
I find it easier to use # # in place of itex, or $ $ in place of tex (without the extra space).
Homework Statement
Prove \lim_{x \to 0} \frac{x}{\sin^2(x) + 1} = 0
Homework Equations
Given below:
The...
Homework Statement
Suppose that limit x> a f(x)= infinity and limit x> a g(x) = c, where c is a real number. Prove each statement.
(a) lim x> a [f(x) + g(x)] = infinity
(b) lim x> a [f(x)g(x)] = infinity if c > 0
(c) lim x> a [f(x)g(x)] = negative infinity if c < 0
Homework Equations...
Homework Statement
Show that ##\lim_{x \to a} f(x) = L## if and only if ##\lim_{x \to 0} f(x+a) = L##
Homework Equations

The Attempt at a Solution
For the forward direction (ie ##1 \Rightarrow 2##), I tried to first assume that 1. holds true (ie ##\forall \epsilon>0, \exists \delta>0...
Homework Statement
The problem is:
f(x) = {ax^2  b / (x  2), x < 2
1/4(x^2)  3c, x >= 2
If f is differentiable at x=2, what are the values of a, b, and c?
In case it is hard to see, I have provided an image
2. Homework Equations
I know that the limit definition of a...
Homework Statement
"Calculate the following limit if it exists. If it does not exist, motivate why.
\displaystyle\lim_{x\rightarrow 0} {\frac{x + x^2 +\sin(3x)}{tan(2x) + 3x}}
Do not use l'Hôpital's rule."
Homework Equations
(1) \sin(a\pm b) = \cos(a)\sin(b)\pm\cos(b)\sin(a)
(2)...
Homework Statement
Find the ## lim _{x> 1+} sqrt(x^23x)2/x+1 ##
Homework Equations
The Attempt at a Solution
I can only solve it using l'hopital rule and would like to know the steps of solving it without using it.
## lim _{x>1+} (2x3)/1= 5/4 ##
Hi,
Suppose you want to prove x  ax + a < \epsilon
You know
x  a < (2a + 1)
You need to prove
x + a < \frac{\epsilon}{2a + 1}
So that
x  ax + a < \epsilon
Why does Michael Spivak do this:
He says you have to prove > x + a < min(1, \frac{\epsilon}{2a + 1}) in...