Short and Sweet Calculus

## 2.3Infinite limits

Consider the function $$f$$ defined by the equation $f(x)=\frac{1}{(x-1)^{2}}$ The graph of this function is illustrated in Figure [fig:ch4-inf-limit-1].

Note that $$f$$ is not defined at $$x=1$$ (division by zero is undefined), but let’s consider the values of $$f$$ when $$x$$ is close to $$1$$. Letting $$x$$ approach $$1$$ from both sides, the corresponding values of $$f$$ are given in the following table.

From this table, we see that as $$x$$ gets closer and closer to $$1$$ but never quite equal to $$1$$, $$f(x)$$ becomes indefinitely large; in other words, we can make $$f(x)$$ as large as we desire if we take $$x$$ close enough to $$1$$.

To express that $$f(x)$$ increases without bound as $$x$$ approaches $$1$$, we write $\lim_{x\to1}\frac{1}{(x-1)^{2}}=+\infty$ Instead of $$+\infty$$, we may simply write $$\infty$$.

• The equals sign does not mean that the limit exists. Note that $$+\infty$$ is not a number; it is just a symbol for indicating that $$f(x)$$ indefinitely increases.

Description: Let $$f$$ be a function defined on both sides of $$a$$, except possibly at $$a$$ itself. If we can make $$f(x)$$ as large as we wish by taking $$x$$ sufficiently close to $$a$$ but not equal to $$a$$, then we say $$f(x)$$ approaches $$\infty$$ as $$x$$ approaches $$a$$ (or the limit of $$f(x)$$, as $$x$$ approaches $$a$$, is positive infinity and write $\lim_{x\to a}f(x)=+\infty$

Now consider the function $$g$$ defined by the equation $g(x)=-\frac{1}{(x-1)^{2}}.$ The graph of $$g(x)$$ is shown in Figure [fig:ch4-inf-limit-4]. We notice that $$g(x)=-f(x)$$ and as $$x$$ approaches $$1$$ from either side, $$g(x)$$ decreases without bound. In this case we write $\lim_{x\to1}g(x)=-\infty.$

Description: Let $$g$$ be a function defined on both sides of $$a$$, except possibly at $$a$$ itself. If we can make $$g(x)$$ as large negative as we wish by taking $$x$$ sufficiently close to $$a$$ but not equal to $$a$$, then we say $$g(x)$$ approaches $$-\infty$$ as $$x$$ approaches $$a$$ (or the limit of $$g(x)$$ is as $$x$$ approaches $$a$$ is infinity and write $\lim_{x\to a}g(x)=-\infty.$

One-sided limits can be also defined accordingly. For example, consider the function $$h(x)$$ defined by the equation $h(x)=\frac{1}{x-2}.$ The graph of this function is represented in Figure [fig:Ch4-inf-limit-5]. In this case, we write $\lim_{x\to2^{-}}h(x)=-\infty,\qquad\text{and}\qquad\lim_{x\to2^{+}}h(x)=+\infty.$