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Calculus Fundamentals

Limits Intuition

In this quiz, we are going to develop some intuition behind the “limiting process.” To do this:

  • We will first look at an example with a mass on a spring.
  • We will then use that example to define the more general problem of limit calculation.
  • From there, we will look at some weird function behavior and use those examples to develop some intuition behind what a limit is.
               
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Limits Intuition

Let us look at an example similar to the distancetime\frac{\text{distance}}{\text{time}} example we saw in the last quiz.

Let x(t)=sin⁡(t)x(t) = \sin(t) model the position of a mass on a spring. Suppose we want its “instantaneous velocity” at the time t=0.t = 0.

To do this, let us look at the average velocity on the interval [0,t][0, t] and examine what happens as we let tt approach 0, as in the last quiz.

Which of the following expressions involving tt correctly models the average velocity change in positionchange in time\frac{\text{change in position}}{\text{change in time}} of the mass on the interval [0,t]?[0, t]?

               
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Limits Intuition

The average velocity of a mass on a spring up to a time tt is sin⁡(t)t.\frac{\sin(t)}{t}.

Similar to what we saw in the previous quiz, if we plug in 0 for tt, the expression evaluates to the 00\frac{0}{0} indeterminate form, so we can't immediately find the value of the instantaneous velocity.

We can, however, look at the value that sin⁡(t)t\frac{\sin(t)}{t} approaches as tt approaches 0.

Using the code environment and graph below, investigate what happens to sin⁡(t)t\frac{\sin(t)}{t} as tt gets closer and closer to zero.

Try, for example, t=1t = 1, t=0.1t = 0.1, and t=0.01t = 0.01. What seems to be happening to sin⁡(t)t?\frac{\sin(t)}{t}?

import math

#change the list below to whatever t values you wish to see evaluated.
#Make sure you keep it in list format (include [] around your list!)
t_vals = [0, 0.5, 1, 1.5, 2]   
 
def velocity(t): #the function we want to see what happens as x approaches 0
	return (math.sin(t)) / (t) 

for t in t_vals:
	print("The value of t: ", t)
	try:
		print("Velocity" , round(velocity(t), 7))
	except  ZeroDivisionError: 
		print("Error Cannot Divide By Zero!")
Python 3
You need to be connected to run code

               
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Limits Intuition

We saw that as tt gets closer and closer to 0, the expression sin⁡(t)t\frac{\sin(t)}{t} gets closer and closer to 1.

We'll have a lot more to say about the details, but this is the basic idea behind saying “the limit as tt approaches 0 of sin⁡(t)t\frac{\sin(t)}{t} is 1.” Symbolically, we write

lim⁡t→0sin⁡(t)t=1.\lim_{t \to 0}\frac{\sin(t)}{t} = 1.

So far, we have worked with limits in relation to a function that models velocity. However, there is no reason we can't generalize to all kinds of functions. This generalization allows us to define the limit problem.

Limit Problem: How can we determine whether or not a general function f(x)f(x) approaches a specific value at x=a?x = a? And if it does, what is this value?

               
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Limits Intuition

For most functions, limit computation is fairly straightforward, but there are a lot of interesting exceptions with weird behavior. Let's analyze one of them so we can get a better understanding of what a limit is.

Suppose we want to determine how the function sin⁡(1x)\sin\left(\frac{1}{x}\right) behaves as xx approaches 0. In other words, we wish to determine

lim⁡x→0sin⁡(1x). \lim_{x \to 0} \sin\left(\frac{1}{x}\right).

Does the function approach a specific value near x=0?x = 0? If so, what value is it?

Use the code environment and graph below to help out. 

import math

#change the list below to whatever x values you wish to see evaluated.
#Make sure you keep it in list format (include [] around your list!)
x_vals = [0, 0.5, 1, 1.5, 2]   
 
def func(x): #the function we want to see what happens as x approaches 0
	return (math.sin(1/x))

for x in x_vals:
	print("The value of x: ", x)
	try:
		print("Value" , round(func(x), 7))
	except  ZeroDivisionError: 
		print("Error Cannot Divide By Zero!")
Python 3
You need to be connected to run code

               
Show explanation

Limits Intuition

The function sin⁡(1x)\sin\left(\frac{1}{x}\right) doesn't seem to approach any particular value as xx approaches 0. When this happens, we say that lim⁡x→0sin⁡(1x)\lim\limits_{x \to 0} \sin\left(\frac{1}{x}\right) does not exist.

Which of these arguments best explains this behavior?

A. \hspace{1mm} As xx gets close to 0, sin⁡(x)\sin(x) also gets close to 0. Therefore, when we take the reciprocal of sin⁡(x)\sin(x), we get something that gets bigger and bigger, so it doesn't approach any particular number.

B. \hspace{1mm} As xx gets close to 0, sin⁡(x)\sin(x) oscillates between 0 and 1, and so does the reciprocal, so it never gets close to any particular number.

C. \hspace{1mm} When you actually plug in x=0,x=0, you get sin⁡(10),\sin\left(\frac{1}{0}\right), which is undefined, so the limit does not exist.

D. \hspace{1mm} As xx gets close to 0, 1x\frac{1}{x} gets larger and larger. Since sin⁡\sin oscillates forever between -1 and 1 as its input gets large, the final result doesn't approach any particular value.

               
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Limits Intuition

Unusual oscillatory behavior can lead to a non-existent limit. Is this pattern true in general?

Let's investigate another oddball example of similar type: lim⁡x→0  xsin⁡(1x).\lim\limits_{x \to 0} \; x \sin\left(\frac{1}{x}\right).

What happens to xsin⁡(1x)x \sin\left(\frac{1}{x}\right) as xx approaches 0?

Use the code environment and graph visualization below to help out.

A. \hspace{1mm} It gets closer and closer to 0, because even though sin⁡(1x)\sin\left(\frac{1}{x}\right) is doing crazy things as xx gets small, we're also multiplying by xx and that's going to 0.

B. \hspace{1mm} It doesn't approach any particular number, because as we saw in the last question, the sin⁡(1x)\sin\left(\frac{1}{x}\right) term oscillates between -1 and 1 as xx gets small.

C. \hspace{1mm} It gets closer and closer to 1.

import math

#change the list below to whatever x values you wish to see evaluated.
#Make sure you keep it in list format (include [] around your list!)
x_vals = [0, 0.5, 1, 1.5, 2]   
 
def func(x): #the function we want to see what happens as x approaches 0
	return x * (math.sin(1/x))

for x in x_vals:
	print("The value of x: ", x)
	try:
		print("Value" , round(func(x), 7))
	except  ZeroDivisionError: 
		print("Error Cannot Divide By Zero!")
Python 3
You need to be connected to run code

               
Show explanation

Limits Intuition

To end this quiz, let's recap what we have done so far:

  • We first examined the notion of a limiting process by analyzing the instantaneous velocity of a mass on a spring at a certain time.
  • We then formulated the limit problem for more general functions.
  • Finally, we examined two oddball examples xsin⁡(1x)x\sin\left(\frac{1}{x}\right) and sin⁡(1x)\sin\left(\frac{1}{x}\right) in order to analyze more interesting edge cases and develop intuition on how to compute limits.

While the two examples seen were stranger than most, these functions give insight into the heart of what a limit actually is.

In the next quiz, we will look at some more examples to gain a better understanding of the limit process.

               
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