Limit Calculator

A limit is the value a function approaches as its input gets closer and closer to a point. It does not matter what the function equals at that point, or even whether the function is defined there. What matters is the trend of the nearby values.

Limits sit at the base of calculus. Derivatives are limits of average rates of change, and integrals are limits of sums. When you ask this calculator for a limit, it probes the function at points nearer and nearer to your target and watches where the values settle. If both sides settle on the same number, that number is the limit.

Mathematicians write a limit as lim x->a f(x) = L. Here f(x) is your function, a is the point x approaches, and L is the value the function tends toward. The arrow means approaches, not equals, which is why a limit can exist at a point where the function itself is undefined.

The calculator estimates L by evaluating f(a - h) and f(a + h) for h = 0.1, 0.01, 0.001, and smaller, then checking that the two sequences agree. If the function is continuous at a, direct substitution f(a) gives the same answer, and the calculator confirms it. If substitution fails with 0/0, the nearby values still reveal the limit. For hand-checking logarithms and exponentials in these steps, the scientific calculator keeps the rounding exact.

Find lim x->1 (x^2 - 1)/(x - 1). Direct substitution gives 0/0, which is undefined, so factor the numerator: x^2 - 1 = (x - 1)(x + 1). Cancel the common factor (x - 1) and the function simplifies to x + 1 everywhere except at x = 1 itself.

Now the limit is easy. As x approaches 1, x + 1 approaches 2. The approach table agrees: at x = 0.9999 the value is 1.9999, and at x = 1.0001 the value is 2.0001. The final answer is 2. The function has a hole at x = 1, but the limit fills it.

1. Type your function in the first field, for example (x^2-1)/(x-1). Use ^ for powers and write 2x or 2*x for multiplication.
2. Set the variable letter, usually x, and type the point it approaches, for example 1.
3. Choose two-sided, left-hand, or right-hand from the direction menu.
4. Press Calculate to see the limit and the approach table behind it. Press Reset to clear the fields.

A two-sided limit asks what the function approaches from every direction at once. A one-sided limit asks only from below (x->a-) or only from above (x->a+). The two-sided limit exists only when both one-sided limits exist and agree.

Take 1/x at 0. From the right the values explode upward: 10, 100, 1000, so the right-hand limit is positive infinity. From the left they explode downward: -10, -100, -1000, so the left-hand limit is negative infinity. The sides disagree, so the two-sided limit does not exist. Choose the direction that matches your question, since one-sided questions appear at graph endpoints and jump discontinuities.

The biggest mistake is treating the function value and the limit as the same thing. Plugging x = 1 into (x^2 - 1)/(x - 1) gives 0/0, yet the limit is 2. Always check the nearby behavior, not just the point.

Two smaller traps catch students too. First, log means base 10 here and ln means the natural log, so log(100) is 2 while ln(e) is 1; mixing them up changes every answer. Second, trigonometric functions use radians, so sin(x)/x approaches 1 only when x is in radians. When a remainder step is part of your algebra, the modulo calculator checks the arithmetic, and the compound interest calculator shows the same exponential growth that limits describe at infinity.