MA101.8 Limits

Consider the function $f(x) = \begin{cases} \frac{-x}{2} + 4&\mbox{} x < 2 \\ -3 &\mbox{} x = 2 \\ x-1 &\mbox{} x > 2\end{cases}$

As x gets closer to 2 from above we notice that f(x) gets as close as we like to 1. We say that f has limit 1 as it tends to 2 from above (some people say from the right).

We write $\displaystyle\lim_{x\to 2^+} f(x) = 1$ or, $f(x) \rightarrow 1$ as $x \rightarrow a^+$.

Note that the value of f at x=2 is not relevant, for the concept of limits we do not even need x=2 to be in the domain for f.

In the same spirit,

$\displaystyle\lim_{x\to 2^-} f(x) = 3$.

Consider the function $f(x) = \begin{cases} \frac{1}{x} &\mbox{} x \neq 0 \\ 2 &\mbox{} x = 2\end{cases}$

As we get closer to 0 from above f(x) gets as large as you like, we say f(x) tends towards infinity.

We write, $\displaystyle\lim_{x\to 0^+} f(x) = \infty$, or
$f(x) \rightarrow \infty$ as $x \rightarrow 0^+$.

It does not mean that f(x) gets close to infinity. The value of f at x=0 is again irrelevant.

In the same spirit we have, $\displaystyle\lim_{x\to0^-} f(x) = -\infty$.

If for a function f(x) and a point x=a we have

$\displaystyle\lim_{x\to a^+} f(x) = \lim_{x\to a^-} f(x) = h \in \mathbb{R}$

then we say that f has a limit h as x tends to a, and we write $\displaystyle\lim_{x\to a} f(x) = h$ or $f(x) \rightarrow h$ as $x \rightarrow a$.

Likewise if $\displaystyle\lim_{x\to a^+} f(x) = \lim_{x\to a^-} f(x) = \infty$ we write simply $\displaystyle\lim_{x\to a} f(x) = \infty$.

For example, $\displaystyle\lim_{x\to 0} \frac{1}{x^2} = \infty$.

Consider the function $f(x) = \frac{2x+3}{x-1},x\neq 1$.

Then $f(x) = \frac{2+\frac{3}{x}}{1- \frac{1}{x}}, x \neq 0, x \neq 1$

As x gets large and positive we see that f(x) gets as close to 2 as we like.

We write $\displaystyle\lim_{x\to \infty} f(x) = 2$ or $f(x) \rightarrow 2$ as $x \rightarrow \infty$.

Likewise $\displaystyle\lim_{x\to -\infty} f(x) =2$ or $f(x) \rightarrow 2$ as $x \rightarrow -\infty$.

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