# Limits of summations and integrals-book recomandation

I'm looking for an exercise's book with valid examples of how to solve limitis of summations and limits of definite integrals. Here some examples:

$\lim_{a\to 0} a\int_{-a}^a\frac{cos x}{x^2 + a^2}$

$\lim_{n\to\infty} \frac{1}{ln n} \sum_{k=0}^{n-1} \frac 1k$

$\lim_{n\to\infty} n^2 \sum_{k=0}^{n-1} sin (2\pi\frac kn)$

Thank you!

• Any good calculus book should do it. My personal favorite is Spivak's calculus book. – Joaquin San Jul 11 '17 at 9:18
• your second limit is $1$ – Dr. Sonnhard Graubner Jul 11 '17 at 9:29
• The first is \pi/2 because cosx=1+O(x^2) – pter26 Jul 11 '17 at 9:30
• @Dr.SonnhardGraubner: the first limit is not undefined. It equals $$\lim_{a\to 0}\int_{-1}^{1}\frac{\cos(az)}{z^2+1}\,dz \stackrel{\text{DCT}}{=}\int_{-1}^{1}\frac{dz}{z^2+1}=\color{red}{\frac{\pi}{2}}.$$ – Jack D'Aurizio Jul 12 '17 at 10:51