Limits Notes

Published on: September 29, 2026

This Limits Calculator finds what a function approaches as x moves towards a chosen point, and shows the working line by line. Substituting the point directly is tried first, and if that gives a usable number the limit is simply that. When it gives an indeterminate form instead, the function is simplified or L'Hopital's rule is applied until it resolves. Following the same pattern every time keeps the substitution attempt visible before any rule is reached for.

Step-by-step method

  1. Set up the problem.
  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:

\(\lim_{x \to a} f\left(x\right) = L\)
See Example 1 Hide Example 1

Problem

\(\lim_{x \to 2} \frac{x^{2} - 4}{x - 2}\)

Approach

Step-by-step method

  1. Set up the problem.
  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.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\lim_{x \to 2} \frac{x^{2} - 4}{x - 2}\)

Step 2

Step 2 - Substitute x = a.

In this problem:

\(f\left(2\right) = \frac{0}{0}\)

Step 3

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

In this problem:

\(\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} = \lim_{x \to 2} x + 2 = 4\)

Step 4

Step 4 - State the final limit.

In this problem:

\(\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} = 4\)

Final Answer

\(\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} = 4\)
See Example 2 Hide Example 2

Problem

\(\lim_{x \to 5} 3 x + 1\)

Approach

Step-by-step method

  1. Set up the problem.
  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.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\lim_{x \to 5} 3 x + 1\)

Step 2

Step 2 - Substitute x = a.

In this problem:

\(\lim_{x \to 5} 3 x + 1 = 16\)

Step 3

Step 3 - State the final limit.

In this problem:

\(\lim_{x \to 5} 3 x + 1 = 16\)

Final Answer

\(\lim_{x \to 5} 3 x + 1 = 16\)
See Example 3 Hide Example 3

Problem

\(\lim_{x \to 5} \frac{x^{2} - 25}{x - 5}\)

Approach

Step-by-step method

  1. Set up the problem.
  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.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\lim_{x \to 5} \frac{x^{2} - 25}{x - 5}\)

Step 2

Step 2 - Substitute x = a.

In this problem:

\(f\left(5\right) = \frac{0}{0}\)

Step 3

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

In this problem:

\(\lim_{x \to 5} \frac{x^{2} - 25}{x - 5} = \lim_{x \to 5} x + 5 = 10\)

Step 4

Step 4 - State the final limit.

In this problem:

\(\lim_{x \to 5} \frac{x^{2} - 25}{x - 5} = 10\)

Final Answer

\(\lim_{x \to 5} \frac{x^{2} - 25}{x - 5} = 10\)
See Example 4 Hide Example 4

Problem

\(\lim_{x \to 4} 2 x^{2} - 3\)

Approach

Step-by-step method

  1. Set up the problem.
  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.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\lim_{x \to 4} 2 x^{2} - 3\)

Step 2

Step 2 - Substitute x = a.

In this problem:

\(\lim_{x \to 4} 2 x^{2} - 3 = 29\)

Step 3

Step 3 - State the final limit.

In this problem:

\(\lim_{x \to 4} 2 x^{2} - 3 = 29\)

Final Answer

\(\lim_{x \to 4} 2 x^{2} - 3 = 29\)