Study Guide

In the Real World

In the Real World

Sometimes measurements and results aren't perfect. This is true in science, engineering, and life. How close to 16.9 fluid ounces of Pythagor-ade can a plastic bottle hold in order to allow 16.9 fluid ounces to be printed on the label? There is a certain range allowed by ShmoopCo that is sure to quench calculus induced thirst, but not too much to send a student to the little girls or boys room mid-class. Using limits, we can answer such questions.

One useful aspect of limits is something many calculus classes don't cover, or don't cover much: the error. To understand error, it helps to understand the formal definition of a limit.

Definition. Let f(x) be a function. We say the limit of f(x) as x approaches a is L, written

if for every real number ε > 0 there exists a real number δ > 0 such that
if |x-a| < δ then |f(x)-  L| < δ.

We can think of ε as the error that's allowed in a measurement. If we know the measurement is the limit of some "nice'' function, the definition of limit says that we can choose the error ε that we want to allow and there will be some δ that will guarantee our measurement will have only the allowed amount of error.

  • I Like Abstract Stuff; Why Should I Care?

    If the whole idea of limits has seemed a little too vague, then we're in luck. It's time to pull ourselves out of the calculus confusion. Let's don our prom attire and prepare for the formal, precise definition for limit.

    Definition. Let f(x) be a function. We say the limit of f(x) as x approaches a is L, written

    if for every real number ε > 0 there exists a real number δ > 0 such that
    if |xa| < δ then |f(x) – L| < δ.

    This will make a lot more sense with a picture. Here's the idea: we're claiming that as x "gets closer" to a, f(x) "gets closer" to L, therefore the function would look something like this:

    or perhaps like this, with a hole in the graph:

    If we try to translate this formal definition into English, it's saying "no matter how close we want f(x) to be to L, if we pick x to be close enough to a we'll find the solution."

    The real number ε is how close we want f(x) to be to L. That is, we want all the function values to fall within the shaded band:

    The real number δ is how close x needs to be to a for the function values to fall within the shaded band:

    If x is anywhere between aδ and a + δ, then f(x) will fall between L ε and L + ε.

    Sample Problem

    Let f(x) = 2x. Then

    If we take, as an example, ε = 0.5, how close does x need to be to 1 in order for f(x) = 2x to be within 0.5 of 2?

    Saying we want f(x) to be within 0.5 of 2 means we want f(x) to be in between 1.5 and 2.5 (in the shaded band on the above graph).
    For f(x) to be greater than 1.5 we need x to be greater than 0.75, and for f(x) to be less than 2.5 we need x to be less than 1.25. Drawing this on the graph, we see that if x is within 0.25 of 1 then f(x) will be in the shaded band.

    If we took any other value of ε > 0, we could find some other value of δ > 0 so that |x – 1| < δ would guarantee |f(x) – 2| < ε. For this function, as long as , we're in the clear. Therefore the limit of f(x) as x approaches 1 is equal to 2.

    Sample Problem

    Let

    This function looks like this:

    It's tempting to say that

    However, if we take a tiny value of ε, no matter how small we take δ to be, there will be some troublesome values of f(x). By troublesome we mean points where x is within δ of 1, but f(x) is not within ε of 5. Therefore the limit of f(x) as x approaches 1 can't possibly be 5.

    No matter how small we take δ to be, there will always be some x-values between 1 – δ and δ with f(x) = 3. If ε is anything less than 2, then f(x) won't be within ε of 5.

    That's just how the cookie crumbles.

  • How to Solve a Math Problem

    There are three steps to solving a math problem.

    • Figure out what the problem is asking.
    • Solve the problem.
    • Check the answer.

    Sample Problem

    Graph the function  

    • Figure out what the problem is asking.

    This problem is asking for a lot. While it may seem like it's asking for a picture, the picture needs to show any holes, vertical asymptotes, horizontal asymptotes, and x or y intercepts of the function.

    • Solve the problem.

    First we need to factor the function:

    If we simplify, we find

    Since the term x is removed from the denominator after simplifying, the function has a hole at x = 0. The full coordinates of the hole are . Since the expressions (x – 3) and (x + 3) are still in the denominator after simplifying, there will be vertical asymptotes at 3 and -3.

    Since the degree of the numerator is less than the degree of the denominator, the function will have a horizontal asymptote at 0.

    In terms of the picture, here's what we have now:

    Now we need to figure out the sign of the function f using a number line:

    When x < -3, both (x – 3) and (x + 3) are negative, therefore f is positive.

    When -3 < x < 3 we know (x – 3) is negative and (x + 3) is positive, therefore f is negative.

    When 3 < x both (x – 3) and (x + 3) are positive, therefore f is positive.

    Now we can fill in the number line:

    Now we have enough information to draw the graph. Since f can't change sign on the interval (-∞, -3) or on the interval (3, ∞), it must look like this:

    • Check the answer.

    For a problem like this, it's probably necessary to double check the solution. Look over the graph to make sure everything is labeled. If necessary, make sure the asymptotes are correct and known values of the function are labeled.

    Use a graphing calculator to check that any graph makes sense.