Limits Calculator

This Limits Calculator evaluates the limit of a function as x approaches a point. It tries direct substitution first, then factoring and cancelling for indeterminate forms like 0/0, and finally L'Hopital's rule when needed - showing each stage as its own step.

Step-by-step method

1. Write the limit in standard form.
2. Substitute x = a.
3. If substitution is indeterminate (like 0/0), simplify the function and substitute again.
4. If it is still indeterminate, apply L'Hôpital's rule.
5. State the final limit.

Formula:

Example 1:

Step 1 - Write the limit in standard form.

In this problem: We are finding the limit of \(\frac{x^{2} - 1}{x - 1}\) as \(x\) approaches \(1\).

Step 2 - Substitute x = a.

In this problem: Substituting \(x = 1\) does not give a defined value. This is the indeterminate form \(\tfrac{0}{0}\).

Step 3 - If substitution is indeterminate (like 0/0), simplify the function and substitute again.

In this problem: Factoring and cancelling gives \(x + 1\), and substituting now gives \(2\).

Step 4 - State the final limit.

In this problem: The limit is \(2\).

Final answer:

Example 2:

Step 1 - Write the limit in standard form.

In this problem: We are finding the limit of \(x \left(x + 3\right)\) as \(x\) approaches \(2\).

Step 2 - Substitute x = a.

In this problem: Substituting gives \(10\), which is defined, so the limit is found directly.

Step 3 - State the final limit.

In this problem: The limit is \(10\).

Final answer: