How to Find the Limit of a Sequence: Formulas and Calculator Checks

Find a sequence's limit by analyzing its formula as the integer index n grows. A calculator can suggest a pattern, but evaluating one very large index does not prove convergence or identify an infinite limit.

What a finite limit means

A sequence u_n converges to L if its terms eventually stay as close to L as required. More precisely, for every ε > 0 there is an integer N such that |u_n − L| < ε whenever n ≥ N.

We write u_n → L as n → ∞. A sequence may instead grow without bound or fail to approach any one value.

Useful limits and algebraic laws

• A constant sequence u_n = c has limit c.
• 1/n^k → 0 for every fixed positive integer k.
• qⁿ → 0 when |q| < 1.
• n^k → +∞ for fixed positive integers k, and qⁿ → +∞ when q > 1.

If u_n → a and v_n → b, with both limits finite:

In particular, the difference law gives a − b, not ab. A zero denominator limit prevents direct use of the quotient law; it does not by itself determine the answer.

Worked example: a rational sequence

Consider u_n = (−2n² + n + 2)/(3n⁴ + 5). Divide numerator and denominator by n⁴:

u_n = (−2/n² + 1/n³ + 2/n⁴)/(3 + 5/n⁴).

Every reciprocal-power term tends to zero. The numerator tends to 0 and the denominator to 3, so the limit is 0.

For rational functions of n, comparing the highest powers is often useful. Equal degrees give the ratio of leading coefficients; a smaller numerator degree gives zero. If the numerator degree is larger, analyze the leading-term sign and growth.

Use a Casio calculator as a numerical check

The original screenshots use a Casio fx-580VN X. Menu names and evaluation controls differ between models; consult your own manual.

Casio model illustration	Alternative calculator illustration

Additional calculator images from the original lesson; no purchase is required to follow the algebra above.

1. Enter the sequence expression using a calculator variable such as x in place of n. Check parentheses around the entire numerator and denominator.
2. Use CALC, where available, to evaluate several positive integer values, such as 10, 100, and 1000.
3. Compare the results with your algebraic prediction. For this example, values should be small and negative for sufficiently large indices, approaching zero.
4. If a very large input causes overflow or suspicious rounding, use smaller values or the algebraically simplified expression.

Entering the rational expression.

Evaluating a selected large input.

A tiny displayed number supports a numerical observation; the division-by-highest-power argument establishes the limit.

Why one large input can mislead

For example, (−1)ⁿ alternates between −1 and 1. Evaluating only large even indices can conceal the absence of a limit. Trigonometric expressions also require the intended angle unit and careful analysis; no universal “use 10⁶ instead” rule proves convergence.

A related result: geometric-series sums

A sequence and a series are different. For the series a + aq + aq² + …, the sum is a/(1 − q) when |q| < 1. Negative ratios can alternate, so this condition does not mean every term decreases monotonically.

For calculator operations, see the Casio calculation guide; check model differences. Graphing tools can provide another visual check, but the formula and limit laws remain the justification.