Calculator Inputs
Pick a family, set the direction, then adjust only the fields that match your chosen form.
Formula Used
Rational: For (a·xn)/(b·xm), compare degrees. If n<m, limit is 0. If n=m, limit is a/b. If n>m, the remaining power controls the sign and divergence.
Power or radical ratio: Rewrite radicals as powers, then use xp/xq = xp-q. The exponent gap decides whether the result is 0, finite, or infinite.
Exponential ratio: Convert a·Arx / b·Bsx into a single rate using e(r ln A - s ln B)x. The sign of that rate controls growth or decay.
Logarithm over power: Any positive power of x dominates any power of ln(x) as x → +∞. That is why many such limits become 0.
Power over logarithm: Positive powers of x outgrow powers of ln(x). Negative powers of x usually force the limit to 0.
Bounded trigonometric over power: Since sin(kx) and cos(kx) remain between −1 and 1, dividing by a growing power of x squeezes the limit to 0.
How to Use This Calculator
- Select the expression family that best matches your limit form.
- Choose whether the variable approaches positive or negative infinity.
- Enter the leading coefficients, degrees, exponents, or logarithm settings.
- Press Calculate Limit to place the answer below the header and above the form.
- Read the classification and step list to verify the dominant-term reasoning.
- Use the export buttons to save a compact CSV summary or a PDF report.
Example Data Table
| Family | Direction | Sample Input | Expected Limit | Reason |
|---|---|---|---|---|
| Rational Function | x → +∞ | (3x4)/(2x2) | +∞ | The numerator degree exceeds the denominator degree. |
| Power or Radical Ratio | x → +∞ | x3/2 /(2x) | +∞ | The exponent gap is 0.5, so the numerator dominates. |
| Exponential Ratio | x → +∞ | (5·2x)/(3·30.5x) | +∞ | The net exponential rate is positive. |
| Logarithm over Power | x → +∞ | (ln x)2 /(4x) | 0 | A positive power of x dominates the logarithm. |
| Bounded Trigonometric over Power | x → +∞ | sin(kx)/(5x2) | 0 | A bounded numerator divided by a growing denominator tends to zero. |
FAQs
What does a limit at infinity describe?
It describes how a function behaves as the input grows without bound in the positive or negative direction. It focuses on long-run end behavior rather than nearby values.
Why do leading terms matter most?
At very large magnitudes, the highest-growth terms dominate everything smaller. That is why lower-degree polynomial pieces or weaker rates can usually be ignored safely.
When does a rational limit equal zero?
A rational limit tends to zero when the denominator has the higher degree. The denominator then grows faster than the numerator, shrinking the whole ratio.
Can I use this for radical expressions?
Yes. Rewrite radicals as powers, such as √x = x1/2, then use the power or radical ratio section. That makes exponent comparison straightforward.
Why are some negative-infinity cases marked undefined?
Certain non-integer exponents and logarithms are not real-valued for negative inputs. The calculator flags those cases so you can switch direction or reformulate the problem.
How are exponential ratios handled?
The calculator combines both exponentials into one effective growth rate. If that rate is positive in the chosen direction, the expression diverges; otherwise it decays to zero.
Why does sin(kx)/x go to zero?
Sine remains trapped between −1 and 1, while x keeps growing. A bounded numerator divided by an unbounded denominator is squeezed toward zero.
What do the export buttons save?
The CSV button saves a compact summary of the active result. The PDF button creates a one-page report containing the expression, direction, classification, and steps.