Thread: Questions arising while studying for midterm.

1. Questions arising while studying for midterm.

I have been going through some practice questions online to prepare for my midterm and ran into a few questions I had problems with but for which there where no explanations for solving.

1. Find limit of $({\frac{1+81x^5}{x^5+5}})^\frac{1}{4}$ as $x$ approaches infinity.
I thought all that would matter in this question is the magnitude of x as I couldn't see any reasonable way to simplify the equation. I answered 1, but the answer is actually 3. What can I do to this problem to make it more manageable?

2. For the given function $f(x)$ and values of $L$, $y$ and $v > 0$ find the largest open interval about $y$ on which the inequality $|f(x)-L| < v$ holds. Then determine the largest value for z > 0 such that $0 < |x-y| < z -> |f(x) - L| < v$.
$f(x) = 4x+7$, $L = 35$, $y = 7$, $v = 0.32$

I worked through all of my textbook questions and never saw something worded like this. I did some work but none of my solutions were close to those that are given. I really have no idea where to start with this question. The answers are: The inequality holds for (6.92, 7.08) and the largest value of $z$ is 0.08.

3. $v = \frac{300x - x^3}{4} \frac{dv}{dx} = \frac{300 - 3x^2}{4}$.

This is part of a larger word problem; however, every step I followed up to this point matched the process in the answer book. Whenever I took the derivative of v using the quotient rule I would come up with an ugly looking quadratic instead of this elegant solution. What am I doing wrong?

2. Originally Posted by Vanilla

1. Find limit of $({\frac{1+81x^5}{x^5+5}})^\frac{1}{4}$ as $x$ approaches infinity.
I thought all that would matter in this question is the magnitude of x as I couldn't see any reasonable way to simplify the equation. I answered 1, but the answer is actually 3. What can I do to this problem to make it more manageable?
Divide top and bottom through by $x^5$ inside the outermost brackets.

CB

3. Originally Posted by Vanilla

1. Find limit of $({\frac{1+81x^5}{x^5+5}})^\frac{1}{4}$ as $x$ approaches infinity.
I thought all that would matter in this question is the magnitude of x as I couldn't see any reasonable way to simplify the equation. I answered 1, but the answer is actually 3. What can I do to this problem to make it more manageable?
I have a love/hate relationship with limits.

$\displaystyle\lim_{x \to \infty}\sqrt[4]{{\frac{1+81x^5}{x^5+5}}}$

$\displaystyle=\lim_{x \to \infty}\sqrt[4]{{\frac{(1+81x^5)\times \frac{1}{x^5}}{(x^5+5)\times \frac{1}{x^5}}}}$

$\displaystyle=\lim_{x \to \infty}\sqrt[4]{{\frac{\frac{1}{x^5}+81}{1+\frac{5}{x^5}}}}$

$\displaystyle=\sqrt[4]{{\frac{\frac{1}{\infty^5}+81}{1+\frac{5}{\infty^5 }}}}$

$\displaystyle=\sqrt[4]{{\frac{0+81}{1+0}}}$

$=\sqrt[4]{81} = 3$