CalculateY
Quick Answer
The limit of (x^2 - 1)/(x - 1) as x approaches 1 is 2. The limit of sin(x)/x as x approaches 0 is 1.
Use * for multiplication. Examples: x^2, sin(x)/x, (x^2 - 1)/(x - 1), 1/x
Common Examples
| Input | Result |
|---|---|
| lim x->1 of (x^2 - 1)/(x - 1) | 2 (removable discontinuity) |
| lim x->0 of sin(x)/x | 1 |
| lim x->0 of 1/x | Does not exist (diverges) |
| lim x->3 of x^2 | 9 |
How It Works
Definition
The limit of f(x) as x approaches a is the value that f(x) gets closer to as x gets closer to a:
\[\lim_{x \to a} f(x) = L\]This means for every small distance from L, there is a corresponding small distance from a where f(x) stays within that distance of L.
One-sided limits
- Left-hand limit (\(x \to a^-\)): approaching from values less than a
- Right-hand limit (\(x \to a^+\)): approaching from values greater than a
The two-sided limit exists only if both one-sided limits exist and are equal.
When limits do not exist
A limit does not exist when:
-
The left and right limits are different (e.g., x /x at x = 0: left = -1, right = +1) - The function diverges to infinity (e.g., 1/x^2 at x = 0)
- The function oscillates without settling (e.g., sin(1/x) at x = 0)
Numeric method
This calculator evaluates f(a - h) and f(a + h) for progressively smaller values of h (0.1, 0.01, 0.001, …, 1e-10). If the values converge to the same number from both sides, the limit exists and equals that value. The convergence table shows the function values at each step.
Worked example
For lim x->1 of (x^2 - 1)/(x - 1): Direct substitution gives 0/0 (indeterminate). Factoring: (x+1)(x-1)/(x-1) = x + 1 for x ≠ 1. As x approaches 1, x + 1 approaches 2. The convergence table confirms: f(0.9) = 1.9, f(0.99) = 1.99, f(0.999) = 1.999, …, converging to 2.