Let c be a real number. Let f be defined for x in (c,infinity) and f(x) > 0 for all x in (c,infinity).?

Show that lim as x-%26gt;c of f = infinity if and only if lim as x approaches c of 1/f = 0.

Let c be a real number. Let f be defined for x in (c,infinity) and f(x) %26gt; 0 for all x in (c,infinity).?
(=%26gt;) Suppose lim (x-%26gt;c) f(x) = ∞ and let ε %26gt; 0. Then let N = 1/ε. Since lim (x-%26gt;c) f(x) = ∞ there exists δ %26gt; 0 such that for c %26lt; x %26lt; c+δ, f(x) %26gt; N %26gt; 0. So for c %26lt; x %26lt; c+δ, 1/f(x) %26lt; 1/N = ε. So lim (x-%26gt;c) 1/f(x) = 0.





(%26lt;=) Suppose lim (x-%26gt;c) 1/f(x) = 0 and let N %26gt; 0. Then let ε = 1/N. Since lim (x-%26gt;c) 1/f(x) = 0 there exists δ %26gt; 0 such that for c %26lt; x %26lt; c+δ, |1/f(x)| %26lt; ε. Since f(x) %26gt; 0 for all x %26gt; c, we have 0 %26lt; 1/f(x) %26lt; ε. So f(x) %26gt; 1/ε = N. So lim (x-%26gt;c) f(x) = ∞.