Limits at Infinity
We can use the knowledge of limits to explore 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. 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 $0$, which is why we call it limits at infinity.
Example 1
Find $\displaystyle \lim_{x \rightarrow \infty}\dfrac{1}{x-1}$.
\( \begin{align} \displaystyle \require{AMSsymbols} \require{color} \lim_{x \rightarrow \infty}\dfrac{1}{x-1} &= \lim_{x \rightarrow \infty}\dfrac{\frac{1}{x}}{\frac{x}{x}-\frac{1}{x}} \\ &= \lim_{x \rightarrow \infty}\dfrac{\frac{1}{x}}{1-\frac{1}{x}} \\ &= \dfrac{0}{1-0} &\color{red} \lim_{x \rightarrow \infty}\dfrac{1}{x} = 0 \\ &= 0 \end{align} \)
Example 2
Find $\displaystyle \lim_{x \rightarrow \infty}\dfrac{3x-2}{x+1}$.
\( \begin{align} \displaystyle \require{AMSsymbols} \require{color} \lim_{x \rightarrow \infty}\dfrac{3x-2}{x+1} &= \lim_{x \rightarrow \infty}\dfrac{\frac{3x}{x}-\frac{2}{x}}{\frac{x}{x}+\frac{1}{x}} \\ &= \lim_{x \rightarrow \infty}\dfrac{3-\frac{2}{x}}{1+\frac{1}{x}} \\ &= \dfrac{3-2 \times 0}{1+0} &\color{red} \lim_{x \rightarrow \infty}\dfrac{1}{x} = 0 \\ &= 3 \end{align} \)
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