Limits of Some Very Nice Functions: Substitution

As you may have noticed in the previous example, for some functions f(x) it is possible to calculate the limit as x approaches a simply by substituting x=a into the function and then evaluating f(a), but sometimes this method does not work. The Substitution Theorem uses the following Two Easy Limits and the Main Limit Theorem to partially answer when such a substitution is valid.

Two Easy Limits: \quad \lim\limits_{x \rightarrow a} \mathrm{k}=\mathrm{k} \quad and \quad \lim\limits_{x \rightarrow a} \mathrm{x}=\mathrm{a}.

Substitution Theorem For Polynomial and Rational Functions:

If \quad \mathrm{P}(\mathrm{x}) and \mathrm{Q}(\mathrm{x}) are polynomials and \mathrm{a} is any number, then \lim\limits_{x \rightarrow a} \mathrm{P}(\mathrm{x})=\mathrm{P}(\mathrm{a}) \quad and \lim\limits_{x \rightarrow a} \frac{P(x)}{Q(x)}=\frac{\mathrm{P}(\mathrm{a})}{\mathrm{Q}(\mathrm{a})} \quad if \mathrm{Q}(\mathrm{a}) \neq 0.

The Substitution Theorem says that we can calculate the limits of polynomials and rational functions by substituting as long as the substitution does not result in a division by zero.

Practice 2: Evaluate (a) \lim\limits_{x \rightarrow 2} 5 x^{3}-x^{2}+3     (b) \lim\limits_{x \rightarrow 2} \frac{x^{3}-7 x}{x^{2}+3 x}     (c) \lim\limits_{x \rightarrow 2} \frac{x^{2}-2 x}{x^{2}-x-2}