# Indeterminate forms

### Indeterminate Forms:

Definition: Indeterminate forms refer to specific mathematical expressions that, upon initial inspection, do not provide enough information to determine their value. These forms typically arise when attempting to evaluate limits.

Common Indeterminate Forms:

1. $\frac{0}{0}$ form:

• Occurs when both the numerator and denominator of a fraction approach zero as the variable approaches a certain value.
• Example: ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}$
2. $\frac{\mathrm{\infty }}{\mathrm{\infty }}$ form:

• Arises when both the numerator and denominator of a fraction grow unbounded as the variable approaches a certain value.
• Example: ${\mathrm{lim}}_{x\to \mathrm{\infty }}\frac{{x}^{2}}{{e}^{x}}$
3. $0\cdot \mathrm{\infty }$ form:

• Emerges when a product involves a factor approaching zero multiplied by another factor approaching infinity.
• Example: ${\mathrm{lim}}_{x\to 0}x\cdot \frac{1}{x}$
4. $\mathrm{\infty }-\mathrm{\infty }$ form:

• Occurs when the difference between two expressions both approaching infinity is indeterminate.
• Example: ${\mathrm{lim}}_{x\to \mathrm{\infty }}\left({x}^{2}-x\right)$
5. ${1}^{\mathrm{\infty }}$ form:

• Arises when a function approaches $1$ raised to the power of another function approaching infinity.
• Example: ${\mathrm{lim}}_{x\to 0}\left(1+x{\right)}^{\frac{1}{x}}$

Approaches for Resolving Indeterminate Forms:

1. Simplification Techniques:

• Simplify the expression by factoring, applying algebraic manipulations, or using trigonometric or logarithmic identities to transform the form into a determinate one.
• Example: Factorization, rationalization, or applying trigonometric or logarithmic properties.
2. L'Hôpital's Rule:

• A technique used for functions in indeterminate forms $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$.
• Involves taking the derivative of the numerator and denominator and re-evaluating the limit.
• Example: ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}$
3. Rewriting Functions:

• Expressing functions involving indeterminate forms in terms of exponentials, logarithms, or trigonometric functions to simplify and evaluate the limit.
• Example: Rewrite ${x}^{x}$ as ${e}^{x\mathrm{ln}\left(x\right)}$ to evaluate the limit.