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## An example explains the impotance of assumption in L’Hospital’s Rule

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 30 Exercises 30.6 Solution: Part aNote that $\cos x\sin x\ge -1$ and $\sin x\ge -1$, we have$$\lim_{x\to\infty}f(x)\ge \lim_{x\to\infty}(x-1)=\infty.$$ Hence…

## Find limits of sequences III

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 7 Exercise 7.5 We shall use the following useful formula. $\sqrt{a}-b=\dfrac{a-b^2}{\sqrt{a}+b}$, which is a variation of $(a-b)(a+b)=a^2-b^2$. Solution: Part…

## Sequence of rational numbers has an irrational limit

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 7 Exercise 7.4 Solution: Part aLet $x_n=\dfrac{1}{\sqrt{n^2+1}}$, then it is clear that all $x_n$ are irrational. However, the sequence…

## Not every ideal is prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.34 Solution: (1) First we show that $I$ is an ideal. Let $f,g \in I$; then for…

## Prove that a given quotient ring is not an integral domain

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.17 Solution: (1) Evidently, $$\overline{p(x)} = \overline{-x^2-11x+3},$$ $$\overline{q(x)} = \overline{8x^2 - 13x + 5},$$ \overline{p(x)+q(x)} = \overline{7x^2…

## Not all ideals are prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.9 Solution: First we show that $I$ is an ideal. To That end, let $f,g \in I$.…

## The quaternion group is not a subgroup of Symmetric group for any n less than 8

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.7 Solution: (1) $Q_8$ is a subgroup of $S_8$ via the left regular representation. (2) Now suppose…