Limits at Infinity

We can use the knowledge of limits to explore of functions for extreme values of $x$, which is the limits of infinity.
  • $x \rightarrow \infty$ to mean when $x$ gets as large as we like and positive,
  • $x \rightarrow -\infty$ to mean when $x$ gets as large as we like and negative.
We read:
  • $x \rightarrow \infty$ as $x$ tends to positive infinity
  • $x \rightarrow -\infty$ as $x$ tends to negative infinity
Notice that as $x \rightarrow \infty$, $1 \lt x \lt x^2 \lt x^3 \cdots$ and as $x$ gets very large, the value of $\dfrac{1}{x}$ gets very small. In fact, we can make $\dfrac{1}{x}$ as close to $0$ as we like by making $x$ large enough.
\begin{array}{|c|c|c|c|c|c|c|} \hline x & 1 & 10 & 100 & 1000 & 10000 & 100 \cdots 0 \\ \hline \dfrac{1}{x} & 1 & 0.1 & 0.0 1& 0.001 & 0.0001 & 0.00 \cdots 1 \\ \hline \end{array} $$\displaystyle \lim_{x \rightarrow \infty}\dfrac{1}{x}=0$$ Note that $\dfrac{1}{x}$ never actually reaches $0$, and that is why we call it as limits at infinity.

Example 1

Find $\displaystyle \lim_{x \rightarrow \infty}\dfrac{1}{x-1}$.

Example 2

Find $\displaystyle \lim_{x \rightarrow \infty}\dfrac{3x-2}{x+1}$.