If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Differential Calculus

### Course: Differential Calculus>Unit 1

Lesson 2: Estimating limits from graphs

# Estimating limit values from graphs

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.C (LO)
,
LIM‑1.C.1 (EK)
,
LIM‑1.C.2 (EK)
,
LIM‑1.C.3 (EK)
,
LIM‑1.C.4 (EK)
The best way to start reasoning about limits is using graphs. Learn how we analyze a limit graphically and see cases where a limit doesn't exist.
There's an important difference between the value a function is approaching—what we call the limit—and the value of the function itself. Graphs are a great tool for understanding this difference.
A function is graphed and animated. The x-axis goes from 0 to 3. The graph is a curve that starts at (0, 0.5), moves downward through an open circle at about (2, 0.25). A cursor moves a point on the curve toward the open circle from the left and the right. Values get close to 0.25. At the open circle, the coordinate displays as (2, undefined).
desmos.com to inspect limit, start subscript, x, \to, 2, end subscript, start fraction, x, minus, 2, divided by, x, squared, minus, 4, end fraction
Notice how, as we get closer and closer to x=2 from both the left and the right, we seem to approach y=0.25.
In the example above, we see that the function value is undefined, but the limit value is approximately 0, point, 25.
Just remember that we're dealing with an approximation, not an exact value. We could zoom in further to get a better approximation if we wanted.

## Examples

The examples below highlight interesting cases of using graphs to approximate limits. In some of the examples, the limit value and the function value are equal, and in other examples, they are not.

### Sometimes the limit value equals the function value.

Problem 1
What is a reasonable estimate for limit, start subscript, x, \to, 1, end subscript, g, left parenthesis, x, right parenthesis ?
Function g is graphed. The x-axis goes from negative 8 to 8. The graph consists of 1 curve. The curve starts at about (negative 7, negative 8) and moves upward through a point at x = 1 between y = negative 1 and y = negative 2, closer to y = negative 1. The curve ends in quadrant 1.

### But, sometimes the limit value does not equal the function value.

Whenever you're dealing with a piecewise function, it's possible to get a graph like the one below.
Problem 2
What is a reasonable estimate for limit, start subscript, x, \to, 1, end subscript, g, left parenthesis, x, right parenthesis ?
Function g is graphed. The x-axis goes from negative 8 to 8. The graph consists of a curve and a closed circle. The curve starts at about (negative 8, 6), moves downward to about (negative 3, 3.5), and moves upward through an open circle at x = 1, just above y = 4. The curve ends in quadrant 1. A closed circle is plotted at x = 1, just below y = 2.

Big takeaway: It's possible for the function value to be different from the limit value.

### And just because a function is undefined for some $x$x-value doesn't mean there's no limit.

Holes in graphs happen with rational functions, which become undefined when their denominators are zero. Here's a classic example:
A function is graphed. The x-axis goes from negative 3 to 3. The graph is a U-shaped curve that starts at about (negative 2.5, 4), moves downward to an open circle at (0, 1), moves upward, and ends at about (2.5, 4).
This is the graph of y = x / sin(x). Notice that there's a hole at x = 0 because the function is undefined there.
In this example, the limit appears to be 1 because that's what the y-values seem to be approaching as our x-values get closer and closer to 0. It doesn't matter that the function is undefined at x, equals, 0. The limit still exists.
Here's another problem for you to try:
Problem 3
What is a reasonable estimate for limit, start subscript, x, \to, minus, 4, end subscript, f, left parenthesis, x, right parenthesis ?
Function f is graphed. The x-axis goes from negative 8 to 8. The graph consists of a curve. The curve starts in quadrant 2 and moves downward through an open circle at x = negative 4, just above the grid line for y = 3. The curve ends in quadrant 4.

Reinforcing the key idea: The function value at x, equals, minus, 4 is irrelevant to finding the limit. All that matters is figuring out what the y-values are approaching as we get closer and closer to x, equals, minus, 4.

### On the flip side, when the function is defined for some $x$x-value, that doesn't mean that the limit necessarily exists.

Just like an earlier example, this graph shows the sort of thing that can happen when we're working with piecewise functions. Notice how we're not approaching the same y-value from both sides of x, equals, 3.
Problem 4
What is a reasonable estimate for limit, start subscript, x, \to, 3, end subscript, g, left parenthesis, x, right parenthesis?

Want more practice? Try this exercise.

## Graphing calculators are pretty slick these days.

Graphing calculators like Desmos can give you a feel for what's happening to the y-values as you get closer and closer to a certain x-value. Try using a graphing calculator to estimate these limits:
\begin{aligned} &\displaystyle{\lim_{x \to 0}{\dfrac{x}{\sin(x)}}} \\\\ &\displaystyle{\lim_{x \to 3}{\dfrac{x-3}{x^2-9}}} \end{aligned}
In both cases, the function isn't defined at the x-value we're approaching, but the limit still exists, and we can estimate it.

## Summary questions

Problem 5
Is it always true that limit, start subscript, x, \to, a, end subscript, f, left parenthesis, x, right parenthesis, equals, f, left parenthesis, a, right parenthesis?

Problem 6
Which statement better describes how graphs help us reason about limits?

## Want to join the conversation?

• How do i know what is the limit if the graph has two functions ?
• So, is it safe to say that a limit exist for a function, if the graph of function at a x-value does not break (discontinued) even if it has another actual value at that same x-value point?
• Yes. Limits are all about where it is heading, not the value at the location.
• When I used desmos to graph x/sin(x), x at 0 was defined. Isn't sin (0)=0, therefore making the function undefined at zero? I know I've done something obviously wrong...can you point it out to me? Thanks
• Since you use desmos, you need to zoom in really close to the part of the graph where x=0 to see that x/sin(x) is not defined at x=0. You would not be able so see this if you graph it with a graphing calculator, however, if you press "2nd" and then "Table" you will see "error" at x=0, which means the function is not defined. I hope this helps.
• What about functions like f(x) = (1/x). Do those have limits?
• Yes, it does except when the function f(x) approaches zero. It is undefined at this point; if you take the limit from the right as x+ >>> 0, it will grow without bound in the positive direction, and if you take the limit from the left as x- >>> 0, it will decrease without bound in the negative direction.

However, if you would take the limit of f(x) as x >>> infinity in either the negative or positive directions, the function would approach a value of 0.
So yes, the function 1/x would have a limit in every circumstance except at the point x = 0.
• Does z-axis takes part in Calculus?
• Yep take a look at multivariate calculus
• This is a bit of a broader question. How is limits applied in the real world? For example, derivatives and integrals are crucial to engineering and the sciences. Are limits just something used in pure mathematics or is there a more applicable use?
• limits form the basis of differentiation and integration.

If you want to find what happens at a particular time or will happen in the long run then limits are useful. Such as in time complexity.
• How do you plug in the limit equation into calculator?
• You have to have an advanced calculator to be able to put in a limit, but you can use numbers that get really close to the number you want to find the limit at. Like if you were finding some functions limit as x approached 3 you would do 2.9, 2.999, 2.999999 and then do 3.1, 3.001, 3.000001, this will allow you to find the limit.
• So, I understand the concept of a limit.
As x approaches the notated value, the limit is the f(x) value it approaches.

Would this work with polar notation?
Like, could you have a limit that defines the value of r as ϴ approaches a given number?