1. ### Tangent of function and its limit position

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)): y-y_0=f^{'}(x_0)(x-x_0) The Attempt at a Solution Applying the above equation for tangent of...
2. ### Limit problem SQRT

Homework Statement [/b] Determine if the limit exists as a number, ∞, -∞ or DNE lim x->4- -(2)/(sqrt(4-x)) 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...
3. ### Finding the number of rational values a function can take

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...
4. ### Evaluating a limit of a function of two variables

I want to evaluate \displaystyle\lim_{(x,y)\to(-1,0)}\frac{y^4(x+1)}{|x+1|^3+2|y|^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...
5. ### Calculus: Limits

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...
6. ### Epsilon delta limits if/then language

< 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...
7. ### Could someone please explain what a limit is?

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
8. ### Continuity at a point implies continuity in the neighborhood

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...
9. ### Prove that arctanx has to be less than x

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...
10. ### Limit and continuity question

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...
11. ### A question on limits

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...
12. ### Proving from first principles that a(n)^2 -> 4 if a(n) -> 2.

Homework Statement Let an → 2. Prove from first principles (i.e. give a direct ε-N proof) that an2 → 4. Homework Equations The Attempt at a Solution I have tried considering |an-2|2 and considering that |an2-4| = |(an+2)(an-2)| but I could not get either of these methods to work. Would...
13. ### Proving limits

lim(9-x) as x->4 = 5 I thought I was supposed to do this: 9-4=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...
14. ### Delta and Epsilon Proofs

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< |x-x0| < delta -> |f(x)-L| < E f(x) = 3-2x, x0=3, E=.02 Here is my attempt: Lim (3-2x) as x->3 = -3 -.02 < |3-2x - 3| <.02 -.02 < |-2x| < .02 .01 > x > -.01 -2.99 > x-3...
15. ### When people help they are not allowed to include answers?

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?
16. ### Continuity and intermediate value theorem

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...
17. ### Limits of Differential Equations

Homework Statement I need help finding the limit of the differential equation. (dx/dt) = k(a-x)(b-x) 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...
18. ### Ln(x) less than root(x)

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.
19. ### Sandwich Theorem: changing inequality

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## ##x-7 \leqslant x+7sinx \leqslant x+7## Now my doubt: I want to divide the expression by...
20. ### A question regarding the definition of e

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...
21. ### Calculating limits

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...
22. ### Calc III infinite integration

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...
23. ### Why does the limit(n->∞) sqrt(n)/(sqrt(n)+sqrt(n+1)) equal 1/(1+sqrt(1-1/n))?

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...
24. ### Can anybody check this proof for a Sine limit?

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...
25. ### Limits with the precise definition of a limit

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...
26. ### Definition of a limit of a function confusion

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...
27. ### Calculus 1 help

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...
28. ### Limit involving trigonometry

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)...
29. ### One-sided limit question

Homework Statement Find the ## lim _{x-> -1+} sqrt(x^2-3x)-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+} (2x-3)/|1|= -5/4 ##
30. ### Question of "min" function from Spivak

Hi, Suppose you want to prove |x - a||x + a| < \epsilon You know |x - a| < (2|a| + 1) You need to prove |x + a| < \frac{\epsilon}{2|a| + 1} So that |x - a||x + a| < \epsilon Why does Michael Spivak do this: He says you have to prove --> |x + a| < min(1, \frac{\epsilon}{2|a| + 1}) in...