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Limit Solver

Compute limits of rational, trig, and exponential functions

Select Function Type

Rational Trig sin(x)/x Exponential Polynomial

Rational: (ax+b)/(cx+d) as x approaches p

lim x->(x+)/(x+)

Limit Evaluation Methods

1. Direct substitution: plug in the value

2. Factor/cancel: factor numerator and denominator

3. LHopital: differentiate top and bottom separately

4. Standard limits: sin(x)/x -> 1, (e^x-1)/x -> 1

Limits describe function behavior as x approaches a value. They are the foundation of calculus continuity, derivatives, and integrals. A limit exists when left and right limits are equal and finite.

⚠ Indeterminate forms (0/0, inf/inf) require algebraic manipulation or LHopital rule before evaluating.

What Is a Limit?

A limit is the value a function approaches as the input approaches some value. Direct substitution is the first approach, but indeterminate forms require additional techniques like factoring, rationalizing, or LHopital rule.

Direct Substitution

Plug x=a into f(x). If result is defined, thats the limit. Works for polynomials and continuous functions.

Indeterminate Forms

0/0, inf/inf, 0*inf, inf-inf. Need factoring, rationalizing, or LHopital. Always check limits from both sides.

LHopital Rule

For 0/0 or inf/inf: lim f/g = lim f/g. Differentiate numerator and denominator separately. Repeat if needed.

Standard Limits

lim sin(x)/x = 1, lim (1-cos(x))/x = 0, lim (e^x-1)/x = 1, lim (1+1/n)^n = e. Memorize these key results.

Teaching Example: lim x->2 (x-4)/(x-2). Direct: 0/0 indeterminate. Factor: (x-4)/(x-2) -> no common factor. The limit does not exist (asymptote at x=2).

Applications

Derivatives Continuity Asymptotes Series Integrals

Frequently Asked Questions

How to compute limits?▼

Try direct substitution first. For 0/0, factor or use LHopital. For inf/inf, compare highest degree terms.

What is LHopital rule?▼

For 0/0 or inf/inf forms: differentiate numerator and denominator separately, then take the limit again.

Standard trig limit?▼

lim x->0 sin(x)/x = 1. This is the fundamental trig limit. All other trig limits derive from this.

When does a limit not exist?▼

When left and right limits differ, function goes to infinity, or oscillates without bound.

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