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Limit Calculator guide

The Limit Calculator evaluates one-sided and two-sided limits numerically. It builds an approach table, estimates the limit, and plots the function to visualize how it behaves as it approaches a specific point.

Worked examples

Evaluate a standard limit

Select the function sin(x)/x.
Set the approach point to 0.
Choose "both" for the side.
Click Save to History to calculate.

Result: lim(x→0) sin(x)/x = 1

Although the function is undefined at x=0, as x gets closer to 0 from both sides, the value approaches 1.

Evaluate a one-sided limit

Select the function 1/x.
Set the approach point to 0.
Choose "right" for the side.

Result: lim(x→0⁺) 1/x = +∞

As x approaches 0 from the positive side, the denominator gets infinitely small, pushing the fraction to positive infinity.

Evaluate a limit at infinity

Select the function (1 + 1/x)^x.
Set the approach point to ∞ (infinity).

Result: lim(x→∞) ≈ 2.7183 (e)

This is the classic definition of Euler’s number, e. As x gets very large, the expression converges to approximately 2.71828.

Tips

•The calculator uses a numerical approach. It evaluates the function at points closer and closer to the target.
•If the left and right limits differ, the two-sided limit Does Not Exist (DNE).
•You can type "inf" or "∞" into the approach field to calculate limits at infinity.

Frequently asked questions

What is a limit?

A limit describes the value a function approaches as the input gets closer and closer to a certain point.

What does DNE mean?

Does Not Exist. A limit DNE if the left-hand limit and right-hand limit are different, or if the function oscillates wildly.

What is a one-sided limit?

A limit where you only approach the target point from one direction (either from values smaller than the target, or larger).

How do you evaluate a limit numerically?

By plugging in values extremely close to the target point and observing the trend of the outputs. The calculator does this automatically.

Can a limit exist if the function is undefined at that point?

Yes. The classic example is sin(x)/x at x=0. The function has a "hole" at 0, but the limit as x approaches 0 is 1.

What is a limit at infinity?

It describes the behavior of a function as the input grows infinitely large (or infinitely negative).

What is L’Hôpital’s Rule?

A calculus method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It involves taking the derivative of the numerator and denominator.

What is the Squeeze Theorem?

If a function is trapped between two other functions that both approach the same limit, the middle function must also approach that limit.

What does "diverges to infinity" mean?

It means the function grows without bound. The limit is not a finite number, but rather +∞ or -∞.

How does the step size affect the numerical limit?

A smaller step size evaluates the function closer to the target point, often giving a more accurate numerical approximation.

Why do I see a table of values?

The table shows the left and right approaches. You can see the output values converging to the limit.

Can a function have multiple limits?

A function can have different left and right limits at a discontinuity, but a standard "two-sided limit" only exists if both sides agree.

What is a continuous function?

A function where the limit as x approaches a point equals the actual function value at that point. You can draw it without lifting your pen.

How do I enter infinity?

You can type "inf", "+inf", "-inf", or use the ∞ symbol.

What are the indeterminate forms?

Expressions like 0/0, ∞/∞, 0×∞, ∞-∞, 1^∞, 0^0, and ∞^0. They do not have a defined value without further analysis.

Can limits be used in real life?

Yes. Limits are the foundation of derivatives and integrals, which are used in physics, engineering, and economics to model rates of change.

What is an asymptote?

A line that a graph approaches but never touches. Limits at infinity or at specific points often reveal asymptotes.

Does this tool show the algebraic steps?

No, it evaluates limits numerically and visually. To see algebraic steps, you would need a symbolic algebra system like WolframAlpha.

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