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Limit Calculator

Evaluate classic one-sided and two-sided limits commonly found in calculus courses and exams. Choose from preset limit problems including removable discontinuities, the fundamental limit sin(x)/x, exponential limits, and radical expressions. Each preset applies the correct algebraic simplification to give the exact classroom answer, not a numerical approximation.

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Formula

Depends on preset: factoring for rational functions, standard limits for trig and exponential forms.

Each preset corresponds to a well-known limit technique: factoring and canceling for removable discontinuities like (x^2 - 4)/(x - 2), the squeeze theorem for sin(x)/x as x approaches 0, L'Hopital's rule or Taylor expansion for exponential limits like (e^x - 1)/x, and conjugate multiplication for radical expressions like (sqrt(x + 4) - 2)/x.

Worked Example

Preset: (x^2 - 4)/(x - 2) as x approaches 2. Step 1: Factor numerator: (x - 2)(x + 2) / (x - 2) Step 2: Cancel (x - 2) from numerator and denominator Step 3: Simplified form = x + 2 Step 4: Substitute x = 2: limit = 2 + 2 = 4 The original function is undefined at x = 2 (0/0), but the limit exists and equals 4.

Frequently Asked Questions

Why are the answers exact rather than approximate?

These presets use algebraic simplification, known theorems, and standard limit identities to produce exact classroom answers. Unlike numerical methods that estimate by plugging in values close to the limit point, this approach gives the mathematically rigorous result.

Can I enter my own custom function?

Not in this version. This calculator covers the most common limit problems from calculus courses. For arbitrary expressions, use a computer algebra system (CAS) like Wolfram Alpha, Desmos, or a symbolic math library.

Why is sin(x)/x equal to 1 at x = 0?

The function sin(x)/x is actually undefined at x = 0 (it gives 0/0). However, the limit as x approaches 0 is exactly 1, proven using the squeeze theorem. This is one of the most fundamental limits in calculus.

What is a removable discontinuity?

A removable discontinuity occurs when a function is undefined at a point but the limit exists. For example, (x^2 - 4)/(x - 2) is undefined at x = 2, but after canceling the common factor, the limit is 4. The "hole" in the graph can be "removed" by defining the function value as the limit.

Does this calculator show step-by-step solutions?

The calculator provides the exact answer with a brief justification for each preset. For full step-by-step derivations with intermediate algebra, working through the problem by hand or using a CAS with step display is recommended.

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