1.3 Finding Limits Analytically

In Section 1.1 we explored the concept of the limit without a strict definition, meaning we could only make approximations. In the previous section we gave the definition of the limit and demonstrated how to use it to verify our approximations were correct. Thus far, our method of finding a limit is (1) make a really good approximation either graphically or numerically, and (2) verify our approximation is correct using an $ϵ$-$\delta$ proof.

Recognizing that $ϵ$-$\delta$ proofs are cumbersome, this section gives a series of theorems which allow us to find limits much more quickly and intuitively.

Suppose that $\underset{x\to 2}{lim}f(x)=2$ and $\underset{x\to 2}{lim}g(x)=3$. What is $\underset{x\to 2}{lim}(f(x)+g(x))$? Intuition tells us that the limit should be 5, as we expect limits to behave in a nice way. The following theorem states that already established limits do behave nicely.

We will now prove the Sum Property using the formal definition of a limit from the previous section. We know that $\underset{x\to c}{lim}f(x)=L$ and $\underset{x\to c}{lim}g(x)=K$. We want to show that $\underset{x\to c}{lim}(f(x)+g(x))=L+K$.

• Proof

  
  We must show that given any $ϵ>0$, we can find a $\delta >0$ such that

  We know $\underset{x\to c}{lim}f(x)=L$. So for any ${ϵ}_1>0$, we can find ${\delta }_1>0$ such that if , then . Similarly we know $\underset{x\to c}{lim}g(x)=K$ so for any ${ϵ}_2>0$, we can find ${\delta }_2>0$ such that if , then . We will let both ${ϵ}_1$ and ${ϵ}_2$ be $\frac{ϵ}{2}$. Now, we have a ${\delta }_1>0$ and a ${\delta }_2>0$ such that:

  	and

  We will choose $\delta =\min ({\delta }_1,{\delta }_2)>0$. If , then and . Add the two inequalities together so that

  We will now use the triangle inequality: $\left|A+B\right|\le \left|A\right|+\left|B\right|$.

  Thus , which is what we were trying to show. ∎

The other Basic Limit Properties can be proven in a similar way and are left for the reader. Our next theorem requires a few more conditions.

We apply the theorem to an example.

Part 3 of the previous example demonstrates how the limit of a quadratic polynomial can be determined using the properties of Theorem 1.3.1. Not only that, recognize that

i.e., the limit at 2 was found just by plugging 2 into the function. This holds true for all polynomials, and also for rational functions (which are quotients of polynomials), as stated in the following theorem.

It was likely frustrating in Section 1.2 to do a lot of work to prove that

as it seemed fairly obvious. The previous theorems state that many functions behave in such an “obvious” fashion, as demonstrated by the rational function in Example 1.3.2.

Polynomial and rational functions are not the only functions to behave in such a predictable way. The following theorem gives a list of functions whose behavior is particularly “nice” in terms of limits. In the next section, we will give a formal name to these functions that behave “nicely.”

Many times, we will combine this theorem with Theorems 1.3.1 and 1.3.2. If our expression can be built up from the pieces in those theorems, then we can quickly evaluate the limit.

The section could have been titled “Using Known Limits to Find Unknown Limits.” By knowing certain limits of functions, we can find limits involving sums, products, powers, etc., of these functions. We further the development of such comparative tools with the Squeeze Theorem, a clever and intuitive way to find the value of some limits.

Before stating this theorem formally, suppose we have functions $f$, $g$ and $h$ where $g$ always takes on values between $f$ and $h$; that is, for all $x$ in an interval,

If $f$ and $h$ have the same limit at $c$, and $g$ is always “squeezed” between them, then $g$ must have the same limit as well. That is what the Squeeze Theorem states, as illustrated in Figure 1.3.1.

It can take some work to figure out appropriate functions by which to “squeeze” the given function of which you are trying to evaluate a limit. However, that is generally the only place work is necessary; the theorem makes the “evaluating the limit part” very simple.

We use the Squeeze Theorem in the following example to finally prove that $\underset{x\to 0}{lim}{\displaystyle \frac{\sin x}{x}}=1$.

Two notes about the previous example are worth mentioning. First, one might be discouraged by this application, thinking “I would never have come up with that on my own. This is too hard!” Don’t be discouraged; within this text we will guide you in your use of the Squeeze Theorem. As one gains mathematical maturity, clever proofs like this are easier and easier to create.

Second, this limit tells us more than just that as $x$ approaches 0, $\sin (x)/x$ approaches 1. Both $x$ and $\sin x$ are approaching 0, but the ratio of $x$ and $\sin x$ approaches 1, meaning that they are approaching 0 in essentially the same way. Another way of viewing this is: for small $x$, the functions $y=x$ and $y=\sin x$ are essentially indistinguishable.

We include this special limit, along with three others, in the following theorem.

A short word on how to interpret the latter three limits. We know that as $x$ goes to 0, $\cos x$ goes to 1. So, in the second limit, both the numerator and denominator are approaching 0. However, since the limit is 0, we can interpret this as saying that “$\cos x$ is approaching 1 faster than $x$ is approaching 0.”

In the third limit, inside the parentheses we have an expression that is approaching 1 (though never equaling 1), and we know that 1 raised to any power is still 1. At the same time, the power is growing toward infinity. What happens to a number near 1 raised to a very large power? In this particular case, the result approaches Euler’s number, $e$, approximately $2.718.$

In the fourth limit, we see that as $x\to 0$, $e^x$ approaches 1 “just as fast” as $x\to 0$, resulting in a limit of 1.

Our final theorem for this section will be motivated by the following example.

The key to the above example is that the functions $y=(x^2-1)/(x-1)$ and $y=x+1$ are identical except at $x=1$. Since limits describe a value the function is approaching, not the value the function actually attains, the limits of the two functions are always equal.

The Fundamental Theorem of Algebra tells us that when dealing with a rational function of the form $g(x)/f(x)$ and directly evaluating the limit $\underset{x\to c}{lim}{\displaystyle \frac{g(x)}{f(x)}}$ returns “0/0”, then $(x-c)$ is a factor of both $g(x)$ and $f(x)$. One can then use algebra to factor this term out, divide, then apply Theorem 1.3.7. Some useful algebraic techniques to rewrite functions that return an indeterminate form when evaluating a limit are:

• •

  factoring and dividing out common factors,

• •

  rationalizing the numerator or denominator,

• •

  simplifying the expression, and

• •

  finding a common denominator.

We will demonstrate some of these techniques in the following examples.

Notice that we didn’t distribute the denominator in the second line. Generally speaking, when we are hoping to divide out a factor in a fraction we will need to undo any distributing that we may have prematurely done.

We end this section by revisiting a limit first seen in Section 1.1, a limit of a difference quotient. Let $f(x)=-1.5x^2+11.5x$; we approximated the limit $\underset{h\to 0}{lim}{\displaystyle \frac{f(1+h)-f(1)}{h}}\approx 8.5.$ We formally evaluate this limit in the following example.

This section contains several valuable tools for evaluating limits. One of the main results of this section is Theorem 1.3.4; it states that many functions that we use regularly behave in a very nice, predictable way. In Section 1.6 we give a name to this nice behavior; we label such functions as continuous. Defining that term will require us to look again at what a limit is and what causes limits to not exist.