## University Calculus: Early Transcendentals (3rd Edition)

(a) $\lim_{x\to c^-}f(x)=\infty$ if for every positive real number $B$, there exists a corresponding number $\delta\gt0$ such that for all $x$: $$c-\delta\lt x\lt c\Rightarrow f(x)\gt B$$ (b) $\lim_{x\to c^+}f(x)=-\infty$ if for every negative real number $-B$, there exists a corresponding number $\delta\gt0$ such that for all $x$: $$c\lt x\lt c+\delta\Rightarrow f(x)\lt -B$$ (c) $\lim_{x\to c^-}f(x)=-\infty$ if for every negative real number $-B$, there exists a corresponding number $\delta\gt0$ such that for all $x$: $$c-\delta\lt x\lt c\Rightarrow f(x)\lt -B$$
(a) We know that as $x\to c$, in the definition, we consider the range $$0\lt |x-c|\lt\delta$$ $$-\delta\lt x-c\lt\delta$$ $$c-\delta\lt x\lt c+\delta$$ As stated in the definition of infinite right-hand limit, as $x\to c^+$, we consider the range $c \lt x\lt c+\delta$, here to build a definition of infinite left-hand limit, we consider instead the range $c-\delta\lt x\lt c$. In detail, $\lim_{x\to c^-}f(x)=\infty$ if for every positive real number $B$, there exists a corresponding number $\delta\gt0$ such that for all $x$: $$c-\delta\lt x\lt c\Rightarrow f(x)\gt B$$ (b) We know that in the definition of infinite limit, $\lim_{x\to c}f(x)=-\infty$ means we would have the inequality $$f(x)\lt-B$$ corresponding every negative real number $-B$. So, to build a definition of $\lim_{x\to c^+}f(x)=-\infty$ from $\lim_{x\to c^+}f)(x)=\infty$, we would have to change only the final inequality. In detail, $\lim_{x\to c^+}f(x)=-\infty$ if for every negative real number $-B$, there exists a corresponding number $\delta\gt0$ such that for all $x$: $$c\lt x\lt c+\delta\Rightarrow f(x)\lt -B$$ (c) To build the definition of $\lim_{x\to c^-}f(x)=-\infty$, we would combine the analysis from part (a) and part (b), using the range $c-\delta\lt x\lt c$ with the inequality $f(x)\lt-B$. In detail, $\lim_{x\to c^-}f(x)=-\infty$ if for every negative real number $-B$, there exists a corresponding number $\delta\gt0$ such that for all $x$: $$c-\delta\lt x\lt c\Rightarrow f(x)\lt -B$$