Problem 8
Evaluate the following: \((a) \lim _{x \rightarrow 4} \frac{x^{2}-x-12}{x-4}\) (b) \(\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}\) (c) \(\lim _{x \rightarrow 0} \frac{5^{x}-e^{x}}{x}\) \((d) \lim _{x \rightarrow \infty} \frac{\ln x}{x}\)
Problem 8
Use bordered determinants to check the following functions for quasiconcavity and quasiconvexity: (d) \(z=-x^{2}-y^{2} \quad(x, y>0)\) \((b): z=-(x+1)^{2}-(y+2)^{2} \quad(x, y>0)\)
Problem 9
By use of L'Hôpital's rule, show that (a) \(\lim _{x \rightarrow \infty} \frac{x^{n}}{e^{x}}=0\) (b) \(\lim _{x \rightarrow 0^{+}} x \ln x=0\) (c) \(\lim _{x \rightarrow 0^{+}} x^{x}=1\)
