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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…

## Application of L’Hospital’s Rule

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 30 Exercises 30.6 Solution: Note that $f(x)=\dfrac{e^xf(x)}{e^x}$, hence we can apply L’Hospital’s Rule \begin{align*} \lim_{x\to\infty}f(x) =&\ \lim_{x\to\infty}\frac{e^xf(x)}{e^x}\\ \text{Apply L’Hospital’s…

## Compute limits using L’Hospital’s Rule II

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 30 Exercises 30.2 Solution: Part aWe can apply L’Hospital’s Rule repeatedly to get the answer. Please check the conditions…

## Compute limits using L’Hospital’s Rule I

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 30 Exercises 30.1 Solution: Part aWe have $\lim_{x\to 0}(e^{2x}-\cos x)=1-1=0$ and $\lim_{x\to 0} x=0$. Hence we can apply L’Hospital’s…

## Determine if the limits exist

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.8 Solution: Part aWe have $\lim n^3=\infty$. Part bWe have $\lim(-n^3)=-\infty$. Part cWe have $\lim(-n)^n$ does not…

## Show the limit is zero

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.7 Solution: We have seen that $0\le s_n<\sqrt{\dfrac{2}{n-1}}$. By Sequeeze-Theorem/Exercise 8.5, it suffices to show that \$\lim…

## Finding the limit by solving equation needs convergence

Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.6 Solution: Part a This part is the same as that of Exercise 9.4 and Exercise 9.5.…