Calculus: Art Of Problem Solving (2), (3), (4)

Hint for Problem 1: the answer is $\displaystyle 2n.$ Here are 3 more problems:

$\displaystyle \star \star \star$ __Problem 2:__ Evaluate $\displaystyle \lim_{n\to\infty} \left[\prod_{k=1}^n \binom{n+k}{n}\right]^{n^{-2}}.$

$\displaystyle \star \star$ __Problem 3:__ Evaluate $\displaystyle \lim_{n\to\infty} \frac{1}{n^k} \int_1^b \prod_{j=1}^k \ln(1 + a_jx^n) \ dx$, where $\displaystyle k, \ a_j > 0, \ b > 1$ are given constants.

$\displaystyle \star \star$ __Problem 4:__ For any real number $\displaystyle x>0$ define $\displaystyle f(x)$ to be the number of elements of the set $\displaystyle \{(m,n) \in \mathbb{Z} \times \mathbb{Z}: \ m^2 + n^2 < x \}.$

Find $\displaystyle \lim_{x\to\infty}\frac{f(x)}{x}.$