- How does the calculator find a limit?
- It starts the way you would on an exam: plug the point in. If f(x) is continuous there, that value is the limit and you're done. If substitution gives 0/0 or ∞/∞, it applies L'Hôpital's rule, differentiating the top and the bottom separately, for up to three rounds, and shows each derivative. Forms like 0·∞ are rewritten as a fraction first, and powers like 1^∞ go through e raised to the limit of g·ln f. Every answer is also checked against f(x) evaluated at points closing in on x. If the algebra and the numbers disagree, you're told so instead of getting a guess.
- What does "Doesn't exist" mean, and why would a limit not exist?
- A two-sided limit only exists when f(x) heads to the same value from the left and from the right. 1/x at 0 fails that test: from the left it heads to −∞, from the right to ∞. abs(x)/x jumps from −1 to 1. The other way a limit fails is oscillation: sin(1/x) swings between −1 and 1 faster and faster as x nears 0 and never settles. The result says which of the two happened, and the table shows the evidence.
- What's the difference between a one-sided and a two-sided limit?
- A one-sided limit only looks at x coming from one direction. x → 0⁻ means from the left (x slightly below 0), x → 0⁺ from the right. The two-sided limit exists only when both one-sided limits exist and match. That's why 1/x at 0 has a right-hand limit of ∞ and a left-hand limit of −∞ but no two-sided limit. When f(x) only lives on one side, like sqrt(x) at 0, the calculator returns that side's limit and tells you that's what it did.
- When can I use L'Hôpital's rule?
- Only on the indeterminate forms 0/0 and ∞/∞. Then the limit of f/g equals the limit of f′/g′, as long as that second limit exists. sin(x)/x at 0 is the classic case: the derivatives give cos(x)/1, which is 1 at x = 0. (1-cos(x))/x^2 needs two rounds: sin(x)/(2x) is still 0/0, and cos(x)/2 settles at 1/2. Using it on anything else, like 1/0, gives wrong answers, which is why the steps name the form before each round.
- Why is (1 + 1/x)^x equal to e as x goes to infinity?
- Substituting gives 1^∞, which is indeterminate: the base creeps toward 1 while the exponent grows without bound, and the two effects fight. Take the natural log: the exponent becomes x·ln(1 + 1/x), a 0·∞ form. Rewrite it as ln(1 + 1/x)/(1/x), which is 0/0, and one round of L'Hôpital shows it heads to 1. So the original limit is e^1 = e ≈ 2.718281828. It's the same limit that defines e in compound interest, where interest paid n times a year approaches continuous compounding.
- Why does x·sin(1/x) have a limit of 0 when sin(1/x) doesn't?
- sin(1/x) keeps swinging between −1 and 1 near 0, but multiplying by x squeezes those swings. The product can never be bigger than |x|, and |x| shrinks to 0, so the product has nowhere to go but 0. That's the squeeze theorem. The Getting closer table makes it visible: the values still change sign, but each one is smaller than the last.
- How do I type functions, and what does "We read this as" show?
- Use + - * / ^ and parentheses, with pi, e, sqrt, abs, exp, ln (natural log), log (base 10), sin, cos, tan, sec, csc, cot, asin, acos and atan. You can skip the multiplication sign (2x, 3sin(x), x(x+1)) and the parentheses on simple arguments (sinx). The echo line rewrites your input with every multiplication as · and extra parentheses where precedence matters. Type x^2-1/x-1 and it shows x^2 − (1/x) − 1, which tells you right away that you meant (x^2-1)/(x-1).
- How exact are the answers?
- When the algebra settles a limit, the value is computed from the simplified expression and matched to an exact form: a whole number, a fraction like 1/2, a multiple of π, a square root like √2/2, or a power of e. The decimal is shown alongside to 10 significant figures. When a limit can only be read from the numbers, the answer is labeled Numeric and the table shows the values it came from. Near the point, floating-point subtraction eventually turns into noise, so the table stops before that happens instead of printing misleading rows.
