Limits describe what a function approaches near a point, even where it is undefined, such as sin(x)/x at 0. Type a function and a point to see values closing in from both sides and the resulting limit.
Type f(x).
Enter the value x approaches.
Read the left, right and overall limit.
Existence: lim x→a f(x) exists ⇔ LHL = RHL
A limit describes the value a function approaches as x gets closer and closer to some number a, whether or not the function is defined at a itself. This calculator evaluates the function at x = a − h and x = a + h for smaller and smaller h, from 0.1 down to 0.00001, lays the values out in two tables, and reports the left-hand limit, the right-hand limit and whether they agree.
Limits are the foundation of calculus and the first new idea in the Class 11 calculus chapter. Standard exam questions involve expressions that look like 0/0 at the point, such as (x² − 9)/(x − 3) at 3 or sin x/x at 0. Seeing the numbers close in from both sides makes the idea concrete and gives a quick check on an algebraic answer.
1. Try direct substitution. If f(a) is a finite number and f is continuous there, that is the limit.
2. If you get 0/0, simplify: factorise and cancel, rationalise a surd, or use a standard result.
3. Standard results: sin x/x approaches 1 as x approaches 0; (1 − cos x)/x² approaches 1/2; (eˣ − 1)/x approaches 1; (xⁿ − aⁿ)/(x − a) approaches n aⁿ⁻¹.
4. For a piecewise or absolute-value function, find the left-hand limit and the right-hand limit separately.
5. The limit exists only if both one-sided limits exist and are equal.
6. If the values grow without bound, the limit is infinite and does not exist as a real number.
Formally, the limit of f(x) as x approaches a equals L if f(x) can be made as close to L as you like by taking x close enough to a, but not equal to a. The value at a itself plays no part. That is why (x² − 9)/(x − 3), undefined at 3, still has a limit there: for every x other than 3 it equals x + 3, which approaches 6. The gap in the graph is called a removable discontinuity, because defining f(3) = 6 would fill it.
Some functions approach different values from each side. |x|/x is −1 for every negative x and +1 for every positive x, so near 0 the left-hand limit is −1 and the right-hand limit is +1, and the two-sided limit does not exist. Step functions like the greatest integer function behave the same way at whole numbers. Other functions, such as 1/x² at 0, grow without bound; we write the limit as infinity, which is a description of behaviour rather than a number. A function is continuous at a exactly when the limit exists and equals f(a).
A table of values is evidence, not proof. The calculator declares agreement when the two sides differ by less than about one part in ten thousand at h = 0.00001. That works for well-behaved functions but can be fooled. sin(1/x) oscillates infinitely fast near 0, so sampled values look random. Expressions like (1 − cos x)/x² involve subtracting nearly equal numbers, and rounding can creep in at very small h. For an exam answer, use algebra, standard limits or L'Hôpital's rule, and use the table to confirm.
Divya, in Class 11, is asked to find the limit of (x² − 9)/(x − 3) as x approaches 3. Direct substitution gives 0/0, so she wants to see what the function does near 3 from both sides.
Values from the left, x = a − h: h = 0.1: 5.9 h = 0.01: 5.99 h = 0.001: 5.999 h = 0.0001: 5.9999 h = 0.00001: 5.99999
Values from the right, x = a + h: h = 0.1: 6.1 h = 0.01: 6.01 h = 0.001: 6.001 h = 0.0001: 6.0001 h = 0.00001: 6.00001
Conclusion: Both sides approach 6
Answer: Limit 6; Left-hand limit 5.99999; Right-hand limit 6.00001
Concluding that a limit does not exist just because the function is undefined at the point.
Cancelling a factor and then forgetting that the simplified form differs from the original at that single point.
Checking only one side for piecewise, modulus or greatest-integer functions.
Using degrees for trigonometric limits; sin x/x approaches 1 only when x is in radians.
Reading a finite-looking table as proof when the function actually oscillates or blows up.
Checking Class 11 and Class 12 limit and continuity answers.
Understanding the definition of the derivative as a limit of slopes.
Testing continuity of piecewise functions at their joins.
Exploring behaviour near asymptotes before sketching a graph.
Estimating the long-run value of a formula by entering a very large x.
Is this a proof of the limit?
No, it is a numerical estimate. Use algebra or L'Hôpital's rule for a formal answer.
Can I find limits at infinity?
Enter a very large value of a, such as 1000000, to see the trend.