2.3 The limit laws  (Page 2/15)

Limits of polynomial and rational functions

By now you have probably noticed that, in each of the previous examples, it has been the case that $\underset{x\to a}{\text{lim}}f\left(x\right)=f\left(a\right).$ This is not always true, but it does hold for all polynomials for any choice of a and for all rational functions at all values of a for which the rational function is defined.

To see that this theorem holds, consider the polynomial $p\left(x\right)=c_n{x}^n+c_{n-1}{x}^{n-1}+\cdots +c_{1}x+c_0.$ By applying the sum, constant multiple, and power laws, we end up with

It now follows from the quotient law that if $p\left(x\right)$ and $q\left(x\right)$ are polynomials for which $q\left(a\right)\ne 0,$ then

[link] applies this result.

Additional limit evaluation techniques

As we have seen, we may evaluate easily the limits of polynomials and limits of some (but not all) rational functions by direct substitution. However, as we saw in the introductory section on limits, it is certainly possible for $\underset{x\to a}{\text{lim}}f\left(x\right)$ to exist when $f\left(a\right)$ is undefined. The following observation allows us to evaluate many limits of this type:

If for all $x\ne a,f\left(x\right)=g\left(x\right)$ over some open interval containing a , then $\underset{x\to a}{\text{lim}}f\left(x\right)=\underset{x\to a}{\text{lim}}g\left(x\right).$

To understand this idea better, consider the limit $\underset{x\to 1}{\text{lim}}\frac{x^2-1}{x-1}.$

The function

and the function $g\left(x\right)=x+1$ are identical for all values of $x\ne 1.$ The graphs of these two functions are shown in [link] .

We see that

The limit has the form $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)},$ where $\underset{x\to a}{\text{lim}}f\left(x\right)=0$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=0.$ (In this case, we say that $f(x)\text{/}g(x)$ has the indeterminate form $0\text{/}0\text{.)}$ The following Problem-Solving Strategy provides a general outline for evaluating limits of this type.