Section 2.10 : The Definition of the Limit
6. Use the definition of the limit to prove the following limit.
limShow All Steps Hide All Steps
Start SolutionFirst, let’s just write out what we need to show.
Let N < 0 be any number. Remember that because our limit is going to negative infinity here we need N to be negative. Now, we need to find a number \delta > 0 so that,
\frac{1}{x} < N\hspace{0.5in}{\mbox{whenever}}\hspace{0.5in} - \delta < x - 0 < 0 Show Step 2Let’s do a little rewrite on the first inequality above to get,
\frac{1}{x} < N\hspace{0.25in}\,\, \to \hspace{0.25in}\,\,\,\,\,x > \frac{1}{N}Now, keep in mind that N is negative and so {\textstyle{1 \over N}} is also negative. From this it looks like we can choose \delta = - \frac{1}{N}. Again, because N is negative this makes \delta positive, which we need!
Show Step 3So, let’s see if this works.
We’ll start by assuming that N < 0 is any number and chose \delta = - \frac{1}{N}. We can now assume that,
- \delta < x - 0 < 0\hspace{0.5in} \Rightarrow \hspace{0.5in}\frac{1}{N} < x < 0So, if we start with the second inequality we get,
\begin{align*}x & > \frac{1}{N} & & \\ & \frac{1}{x} < N & & \hspace{0.25in}{\mbox{ rewriting things a little bit}}\end{align*}So, we’ve shown that,
\frac{1}{x} < N\hspace{0.5in}{\mbox{whenever}}\hspace{0.5in}\frac{1}{N} < x < 0and so by the definition of the limit we have just proved that,
\mathop {\lim }\limits_{x \to {0^ - }} \frac{1}{x} = - \infty