How limits are evaluated
First try to substitute the point: if you get a number, for a continuous function that is the limit. If the denominator tends to zero and the numerator to a non-zero number, the fraction tends to infinity, with the sign depending on the side. If you get 0/0, ∞/∞, 0·∞, ∞ − ∞, 1^∞, 0⁰ or ∞⁰, it is an indeterminate form and the expression must be transformed.
Techniques
- 0/0 for polynomials — factor and cancel (x − a).
- ∞/∞ as x → ∞ — divide by the highest power of x; the leading terms decide.
- Roots — multiply by the conjugate to get a difference of squares.
- Equivalent infinitesimals (u → 0): sin u ~ u, tan u ~ u, arcsin u ~ u, arctan u ~ u, ln(1 + u) ~ u, eᵘ − 1 ~ u, aᵘ − 1 ~ u·ln a, 1 − cos u ~ u²/2, (1 + u)ᵏ − 1 ~ k·u. Replace factors only, never terms of a sum.
- The first special limit: sin x / x → 1 as x → 0.
- The second special limit: (1 + 1/x)ˣ → e as x → ∞; in general, for 1^∞, lim u^v = e^(lim v·(u − 1)).
- L'Hôpital's rule: for 0/0 and ∞/∞ the limit of a ratio equals the limit of the ratio of derivatives, if the latter exists. Write 0·∞ as a fraction first; 0⁰ and ∞⁰ via u^v = e^(v·ln u).
When the limit does not exist
A two-sided limit exists only if the left and right limits agree: for 1/x at 0 they are −∞ and +∞, for e^(1/x) they are 0 and +∞. sin(1/x) oscillates between −1 and 1 near zero.
Why there is a numerical check
The calculator always tabulates f(x) near the point. If no analytic technique applies, the answer is numerical and marked with ≈ — no made-up steps.