For $a>0$, the following hold:

(i) $e^{ax}\rightarrow\infty$ as $x\rightarrow\infty$;

(ii) $e^{-ax}\rightarrow 0$ as $x\rightarrow\infty$;

(iii) $x^{a}\rightarrow\infty$ as $x\rightarrow\infty$;

(iv) $x^{a}\rightarrow 0$ as $x\rightarrow 0+$;

(v) $\log x\rightarrow\infty$ as $x\rightarrow\infty$;

(vi) $\log x\rightarrow-\infty$ as $x\rightarrow 0+$;

(vii) $r^{x}\rightarrow\infty$ as $x\rightarrow\infty$ for $r>1$;

(viii) $r^{x}\rightarrow 0$ as $x\rightarrow\infty$ for $0<r<1$;

(ix) ${{\sin x}\over{x}}\rightarrow 1$ as $x\rightarrow 0$;

(x) $\tan^{-1}x\rightarrow\pi/2$ as $x\rightarrow\infty$.