Limit Calculator
Evaluate limits of functions as x approaches a specific point. Supports two-sided limits and one-sided limits (left and right).
What is a limit in calculus?
A limit describes the value a function approaches as the input approaches a particular point. Written as lim(x→a) f(x) = L, it means as x gets closer to a, f(x) gets closer to L. Example: lim(x→2) x^2 = 4. Limits are fundamental to calculus, defining derivatives and integrals.
How do you evaluate limits algebraically?
For simple limits: 1) Direct substitution - plug in the value. 2) Factor and cancel - for 0/0 forms, factor numerator/denominator. 3) Multiply by conjugate - for radical expressions. Example: lim(x→3) (x^2-9)/(x-3) = lim(x→3) (x+3)(x-3)/(x-3) = lim(x→3) (x+3) = 6.
What does it mean when a limit does not exist?
A limit DNE (does not exist) when: 1) Left and right limits differ. 2) Function approaches infinity. 3) Function oscillates infinitely. Example: lim(x→0) 1/x DNE because left limit = -∞, right limit = +∞. lim(x→0) sin(1/x) DNE due to oscillation.
What is the difference between one-sided and two-sided limits?
Two-sided limit lim(x→a) f(x) requires function to approach same value from both sides. One-sided: lim(x→a⁺) approaches from right, lim(x→a⁻) from left. Example: For f(x) = |x|/x at x=0, left limit = -1, right limit = 1, so two-sided limit DNE.
How are limits used in derivatives?
The derivative is defined as a limit: f'(x) = lim(h→0) [f(x+h)-f(x)]/h. This measures instantaneous rate of change. Example: For f(x)=x^2, f'(x) = lim(h→0) [(x+h)^2-x^2]/h = lim(h→0) [2xh+h^2]/h = lim(h→0) (2x+h) = 2x.