Limit Calculator
Estimate the limit of a function as x approaches a point or infinity (two-sided or one-sided).
Details
Use ^ for powers, and sin, cos, ln or sqrt for functions.
You are finding the limit of
lim x → 0 ( sin(x) / x )
Limit
…
This finds what a function approaches as x heads towards a value. Enter the function and the value, and choose whether to approach from both sides or just one.
It returns the limit along with the working, and tells you when the two sides disagree, which is when the limit does not exist.
What is a limit?
A limit asks: as x gets closer and closer to some number, what value does the function head towards?
The crucial part is that it does not care what happens *at* that point. It cares where the function is going. A function can be completely undefined at x = 2 and still have a perfectly good limit there, because you are watching the approach, not the destination.
That distinction is what makes calculus work. Both derivatives and integrals are defined as limits, since both involve squeezing something down towards zero without ever quite reaching it.
Read it as: as x approaches 0, sin(x)/x approaches 1. Putting x = 0 in directly gives 0/0, which is meaningless — yet the limit is a clean 1, because that is where the function is heading.
What to enter
- Your function
- Type it as you would say it: sin(x)/x, (x^2-1)/(x-1), 1/x. Use ^ for powers.
- What does x approach?
- The value x is heading towards. You can also use infinity to see how the function behaves far out.
- Which side?
- Both sides is the normal choice. Left means approaching from below, right means from above. They matter when the function jumps.
The three kinds of limit, and when each matters
- Two-sided
- The default. Requires the function to head to the same value from both directions. If the two sides disagree, the limit does not exist.
- One-sided
- Written with a small + or − on the approach value. Used where a function behaves differently either side of a point, like a step or a price that jumps at a threshold.
- Limit at infinity
- Asks what the function settles down to as x grows without bound. This is what finds horizontal asymptotes.
What this assumes
Trigonometric functions use radians. sin(x)/x only approaches 1 in radians; in degrees it approaches a different number entirely.
The result is found numerically and symbolically, so a function that oscillates forever near the point may not settle on an answer.
How to calculate a limit
Try substituting first. It works far more often than students expect, and you only need the harder methods when it fails.
- a
- The value x is approaching
- L
- The value the function heads towards
Put the number in. If you get a sensible answer, that is the limit and you are finished.
If you get 0/0, factor and cancel. 0/0 is not an answer, it is a signal that something cancels. Factor the top and bottom and remove the common part.
Then substitute again. Once the troublesome factor is gone, putting the number in usually works.
For limits at infinity, compare growth. Divide everything by the highest power of x present. Terms with x underneath shrink to zero and the answer falls out.
See a worked example: why (x² − 1)/(x − 1) approaches 2 as x approaches 1
- Function
- (x² − 1) ÷ (x − 1)
- x approaches
- 1
Substituting 1 gives (1 − 1)/(1 − 1) = 0/0, which tells you nothing yet.
Factor the top: x² − 1 is a difference of squares, so it becomes (x − 1)(x + 1).
Now cancel the (x − 1) that appears top and bottom, leaving just x + 1.
Substitute again: 1 + 1 = 2.
Note the function genuinely does not exist at x = 1, since that would be dividing by zero. The limit is still 2, because that is where it is heading from both sides.
The limit is 2
Frequently asked questions
It means the method failed, not that the answer is zero or one. It is called an indeterminate form: the answer could be anything, so you need more work.
It is usually good news. It almost always signals a factor that cancels, and once you remove it the limit appears.
Most often when the two sides disagree. If the function heads to 3 from the left and 5 from the right, there is no single value it is approaching, so the two-sided limit does not exist.
It also fails if the function shoots off to infinity, or oscillates forever without settling, as sin(1/x) does near zero.
A two-sided limit needs both directions to agree. A one-sided limit only looks one way.
They matter wherever a function jumps: a parking charge that changes on the hour, or a tax band. Approaching from below and from above genuinely give different answers, and both are meaningful.
Yes, and this is the whole idea. (x² − 1)/(x − 1) does not exist at x = 1, because that is division by zero, but its limit there is a perfectly good 2.
A limit describes the approach, never the arrival. That is exactly why it can see past a hole in the graph.
Because for very small angles measured in radians, sin(x) and x are almost identical. At x = 0.01, sin(x) is 0.0099998, so the ratio is a whisker under 1.
The radians part is essential. In degrees the same expression approaches about 0.01745 instead, which is why calculus always uses radians.
Problems people actually run into
Treating 0/0 as an answer
Writing 'the limit is 0/0' or concluding the limit does not exist is the most common error here. 0/0 is a prompt to keep working, not a result.
The usual fix is factoring and cancelling. Where that will not work, multiplying by the conjugate handles most square-root cases.
Assuming the limit equals the function's value
They match for smooth functions, which is what makes those functions continuous. They do not have to match in general.
A graph can have a hole at a point, or be defined at a completely different value there, and the limit still describes the approach rather than that isolated point.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: September 3, 2026