Give rigorous formulations of the following statements. i) $f(x) \to \infty $ for $x \to \infty $
ii) $f(x) \to -\infty $ for x $ \to \infty $
iii) $f(x) \to \infty $ for $x \to -\infty $
iv) $f(x) \to -\infty $ for $x \to -\infty $
v) $f(x) \to \infty $ for $x \to a$
vi) $f(x) \to -\infty $ for $x \to a$ Are my formulations correct?
i) $ \forall b \in \R, \exists c \in \IR : x > c \Rightarrow f(x)> b$
ii) $ \forall b \in \IR, \exists c \in \IR : x > c \Rightarrow f(x)< b$
iii)$ \forall $ b $ \in \IR, \exists $ c $ \in \IR $ : x < c $ \Rightarrow $ f(x)> b
iv) $ \forall $ b $ \in \IR, \exists $ c $ \in \IR $ : x < c $ \Rightarrow $ f(x)< b
v) $ \forall $ b $ \in \IR, \exists $ a $ \in \IR $ : x > a$ \Rightarrow $ f(x)>b
vi)$ \forall $ b $ \in \IR, \exists $ a $ \in \IR $ : x > a $ \Rightarrow $ f(x)< b ????? I am not sure about my formulations so please tell me if I need to correct them.... thanks
v) should be $\forall b\in{\bf{R}}$, $\exists\delta>0$, $\forall x$, $0<|x-a|<\delta\rightarrow f(x)>b$, similar for vi).