Hello everyone,
I'm posting here since I'm only having trouble with an intermediate step in proving that
\sqrt{x} \text{ is uniformly continuous on } [0, \infty] .
By definition, |x - x_0| < ε^2 \Longleftrightarrow -ε^2 < x - x_0 < ε^2 \Longleftrightarrow -ε^2 + x_0 < x < ε^2 + x_0
1...
Thank you for your responses, Bohrok and HallsofIvy.
@HallsofIvy: Thanks for your clarification on conjugates.
How would you define a real conjugate then? Are two terms x and y conjugates of each other if and only x \times y are of degree 1 and do not have any fractional exponents?
Homework Statement
Evaluate the limit, WITHOUT using l'Hôpital's rule:
\lim_{x \rightarrow -1} \frac{x^{1/3} + 1}{x^{1/5} + 1}
Homework Equations
The Attempt at a Solution
I tried to use the conjugate method which does not produce a useful outcome:
\underset{x\to...
Homework Statement
[B]Evaluate \underset{x\to 0}{\mathop{\lim }}\,\frac{\sec \frac{x}{2}-1}{x\sin x} , WITHOUT using l'Hôpital's rule.
Homework Equations
The Attempt at a Solution
Hello there,
I tried to evaluate this limit using two different approaches, both of which still...
Adminstrator: This is a double post, so please feel free to delete this one.
The relevant post is "Limit of a Trigonometric Function (Involved Problem)".
Homework Statement
Evaluate \underset{x\to 0}{\mathop{\lim }}\,\frac{\sec \frac{x}{2}-1}{x\sin x}
Homework Equations
The Attempt at...
Thanks for your response, Hurkyl.
Proof that \lim_{x \rightarrow 0} \frac{1}{x} does not exist:
\lim_{x \rightarrow 0^{-}} \frac{1}{x} = \frac{1}{0^{-}} = -\infty
\lim_{x \rightarrow 0^{+}} \frac{1}{x} = \frac{1}{0^{+}} = \infty
Since the right- and left-sided limits differ and...
Hello there,
I would like to learn how I can use the formal definition of a limit to prove that a limit does not exist. Unfortunately, my textbook (by Salas) does not offer any worked examples involving the following type of limit so I am not sure what to do. I write below that delta = 1 would...
Thanks for your reply, Dick.
I have:
\vec{u} = |\vec{u}| \cos A1 \hat{i} + |\vec{u}| \cos B1 \hat{j} + |\vec{u}| \cos Y1 \hat{k}
\vec{v} = |\vec{v}| \cos A2 \hat{i} + |\vec{v}| \cos B2 \hat{j} + |\vec{v}| \cos Y2 \hat{k}
If I dot these two expressions on the RS, I do not see...
Thank you for your response.
Would you mind elaborating on the proof?
I thought that the direction cosines themselves were the unit vectors, so how would \cos A1 \cos A2 = 0 ? Shouldn't the dot product of these direction cosines = 0?
Hello everyone,
Thank you in advance for your help!
---
Homework Statement
10. A vector \vec{u} with direction angles A1, B1, and Y1, is perpendicular to a vector \vec{v} with direction angles A2, B2, and Y2. Prove that:
\cos A1 \cos B2 + \cos B1 \cos B2 + \cos Y1 \cos Y2 = 0.
---...
Hello there:
Here is my solution, which matches your textbook's.
y = \cos(a^3 + x^3)
Let: u = a^3 + x^3
The Chain Rule: h'(x) = f'(g(x)) \times g'(x)
Breaking the original equation down:
f(x) = \cos u , f'(x) = -\sin u
g(x) = u = a^3 + x^3 , g'(x) = 3x^2 (From the Product Rule)...