The Limit Laws

We can prove that a limit takes on a given value using the epsilon-delta definition, but doing so proves cumbersome, even for simple functions. What's worse is the fact that doing so requires you know what the limit value should be before you start! Many times, it is obvious if we have some familiarity with the graph of the function. However, at other times it may not be so clear. For example, you probably have not graphed the following function before. $$f(x)=x-\sqrt{x}+\frac{\sin^2(3x)}{2x^2}$$ Without any ability to find out what the graph looked like, could you still determine the value of the following limit? $$\lim_{x \rightarrow 0} \left[ x-\sqrt{x}+\frac{\sin^2(3x)}{2x^2} \right]$$ To address this concern, we can take a lesson from the manufacturing industry. In manufacturing, one takes simple materials and combines them to form something more complex.

Similarly, we can start with simple functions -- and if we understand how to find limits of these simple functions, as well as how the limits of combinations of functions relate to the limits and values of the original functions -- then we should be able to find limits of arbitrarily complex combinations of these simpler functions.

The function in the example above is a prime example -- it is "built" from simpler functions ($\sin x$, $3x$, $x^2$, $\sqrt{x}$, etc...) which are then combined through the operations of addition, subtraction, multiplication, division, and composition.

Of course to find the limit of such a monster, we still need to know how to find limits for the simple functions involved, and we know how limits are affected by combining functions. Fortunately, we can use the epsilon-delta definition to establish these things, which collectively are commonly referred to as the "Limit Laws".

To be specific...

Using the epsilon-delta definition of a limit, we can prove all of the results below -- each one of which involves the limit of "simple function":

  1. $\displaystyle{\lim_{x \rightarrow c} \, a = a}$ (for any constant $a$)

  2. $\displaystyle{\lim_{x \rightarrow c} \, x = c}$

  3. $\displaystyle{\lim_{x \rightarrow c} \, \sqrt[n]{x} = \sqrt[n]{c}}$ (for any positive integer $n$, assuming $c>0$ when $n$ is even). Also, $\displaystyle{\lim_{x \rightarrow 0^+} \, \sqrt[n]{x} = 0}$ when $n$ is even.

  4. $\displaystyle{\lim_{x \rightarrow c} \, |x| = |c|}$

  5. $\displaystyle{\lim_{x \rightarrow c} \, \sin(x) = \sin(c)}$

  6. $\displaystyle{\lim_{x \rightarrow c} \, \cos(x) = \cos(c)}$

  7. $\displaystyle{\lim_{x \rightarrow c} \, \ln(x) = \ln(c)}$ (for any $c \gt 0$ -- otherwise the limit fails to exist)

  8. $\displaystyle{\lim_{x \rightarrow c} \, e^x = e^c}$

As straight-forward as the above appear, the epsilon-delta proofs of some of these results are not at all obvious (and sometimes, quite complicated). If you are curious about the details, you might try finding a calculus textbook and looking in the appendix.

We can also prove the following results regarding limits of "combinations of functions" using the epsilon-delta definition:

Assuming that $\lim_{x \rightarrow c} f(x)$ and $\lim_{x \rightarrow c} g(x)$ exist:
  1. $\displaystyle{\lim_{x \rightarrow c} \, [f(x) + g(x)] = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c} g(x)}$

  2. $\displaystyle{\lim_{x \rightarrow c} \, [f(x) - g(x)] = \lim_{x \rightarrow c} f(x) - \lim_{x \rightarrow c} g(x)}$

  3. $\displaystyle{\lim_{x \rightarrow c} \, [f(x)g(x)] = \lim_{x \rightarrow c} f(x) \cdot \lim_{x \rightarrow c} g(x)}$

  4. $\displaystyle{\lim_{x \rightarrow c} \, \frac{f(x)}{g(x)} = \frac{\displaystyle{\lim_{x \rightarrow c} \, f(x)}}{\displaystyle{\lim_{x \rightarrow c} \, g(x)}} \quad \textrm{, if } \lim_{x \rightarrow c} \, g(x) \neq 0}$

  5. If $\displaystyle{\lim_{x \rightarrow c} \, g(x) = b}$ and $\displaystyle{\lim_{x \rightarrow b} \, f(x) = f(b)}$, then $\displaystyle{\lim_{x \rightarrow c} \, f(g(x)) = f(\lim_{x \rightarrow c} \, g(x))}$

Also, as a quick - but very useful - direct consequence of this last result, note that $$\textrm{If } \lim_{x \rightarrow c} \, f(x) = f(c) \textrm{ for all real values $c$, then $\lim_{x \rightarrow a} \, f(g(x)) = f(\lim_{x \rightarrow a} \, g(x))$}$$

We can prove other very useful theorems in the same vein, by combining some of the above results. For example, combining the first "simple function" result above and the third "combination of functions" result, we see that $$\displaystyle{\lim_{x \rightarrow c} \, [a f(x)] = a \lim_{x \rightarrow c} \, g(x)} \quad \textrm{(for any constant $a$)}$$ Similarly, we can combine the second "simple function" result with the third "combination of functions" result repeatedly, to find $$\displaystyle{\lim_{x \rightarrow c} \, x^n = c^n} \quad \textrm{(for any positive integer $n$)}$$ Taking things one step further, we can put together the previous two results with the first two "combination of functions" results to obtain (for any positive integer $n$ and constants $a_1,a_2, ... , a_n$): $$\displaystyle{\lim_{x \rightarrow c} \, (a_n x^n + \cdots + a_2 x^2 + a_1 x + a_0) = a_n c^n + \cdots + a_2 c^2 + a_1 c + a_0}$$ Of course, we can say this much more simply in the following manner: $$\displaystyle{\lim_{x \rightarrow c} \, p(x) = p(c) \quad \textrm{whenever $p(x)$ is a polynomial function}}$$

Recall, that we can combine functions in ways other than addition, subtraction, multiplication, division, and composition, in order to form new functions. One very common way to do this is through piece-wise defined functions. At other times, we may not be able to "build" the function in question from simpler ones, but we may be able to "bound" it between simpler ones. To these ends, the following results prove very useful:

  • If $f(x)=g(x)$ everywhere except at $x=c$, then $\displaystyle{\lim_{x \rightarrow c} f(x) = \lim_{x \rightarrow c} g(x)}$

  • $\displaystyle{\lim_{x \rightarrow c} \, f(x) = L \textrm{ if and only if } \lim_{x \rightarrow c^-} \, f(x) = \lim_{x \rightarrow c^+} \, f(x)}$

  • The Squeeze Theorem:
    If $f(x) \le g(x) \le h(x)$ when $x$ is sufficiently close to $c$ (except possibly at $x=c$), and $$\lim_{x \rightarrow c} \, f(x) = \lim_{x \rightarrow c} \, h(x) = L$$ then $$\lim_{x \rightarrow c} \, g(x) = L$$

(Again, all of these results can be proven true by appealing to the epsilon-delta definition of a limit.)

Incidently, we can add one more "simple function" limit to our list as a direct (but not necessarily obvious) result of the Squeeze Theorem mentioned above: $$\lim_{x \rightarrow 0} \, \frac{\sin x}{x} = 1$$ This may seem like a peculiar result, whose application would be fairly limited -- but amazingly, this will get used way more often than you might think!