### Shortcut Methods

**Shortcut Methods and Tricks to Solve Numericals:**

**JEE Mains:**

**Factoring**: Factor the numerator and denominator to simplify the expression. If the expression becomes (\frac{0}{0}), apply L’Hôpital’s rule or other appropriate techniques.**Cancellation**: Look for common factors that can be canceled between the numerator and denominator. This can sometimes simplify the expression significantly.**Substitution**: If the limit involves an indeterminate form of the type (\frac{0}{0}), (\frac{\infty}{\infty}), or (0 \cdot \infty), try substituting the limit value directly into the expression.

**CBSE Board Exams:**

**Direct Substitution**: For simple expressions, directly substitute the limit value into the function to find the limit.**Graphical Analysis**: Use a graphing calculator or plot the graph of the function to visualize the behavior of the function near the limit point. This can help determine whether the limit exists and its value.**L’Hôpital’s Rule**: If direct substitution results in an indeterminate form ((\frac{0}{0}) or (\frac{\infty}{\infty})), apply L’Hôpital’s rule to find the limit by taking the derivative of the numerator and denominator.