Evaluation of Limits

Objective: The evaluation of limits involves determining the value that a function approaches as the independent variable approaches a particular value or infinity.

Methods of Evaluating Limits:

1. Direct Substitution:

• If substituting the value of $x$ into the function gives a finite value and doesn't lead to division by zero, then the limit is the value obtained by direct substitution.
• Example: ${\mathrm{lim}}_{x\to 3}\left(2x-1\right)=2\left(3\right)-1=5$
2. Factoring and Simplification:

• For functions involving polynomials or rational expressions, factorizing and simplifying can help evaluate limits by canceling common terms.
• Example: ${\mathrm{lim}}_{x\to 2}\frac{{x}^{2}-4}{x-2}={\mathrm{lim}}_{x\to 2}\frac{\left(x+2\right)\left(x-2\right)}{x-2}={\mathrm{lim}}_{x\to 2}\left(x+2\right)=4$
3. Conjugate Method:

• Useful for limits involving square roots or complex conjugates; multiplying by the conjugate of the denominator helps eliminate radicals.
• Example: ${\mathrm{lim}}_{x\to 0}\frac{\sqrt{x+9}-3}{x}={\mathrm{lim}}_{x\to 0}\frac{\left(\sqrt{x+9}-3\right)\left(\sqrt{x+9}+3\right)}{x\left(\sqrt{x+9}+3\right)}={\mathrm{lim}}_{x\to 0}\frac{x}{x\left(\sqrt{x+9}+3\right)}=\frac{1}{6}$
1. L'Hôpital's Rule:

• Used for limits involving indeterminate forms such as $\frac{0}{0}$ or $\frac{\mathrm{\infty }}{\mathrm{\infty }}$. It applies to the ratio of derivatives of two functions.
• Example: ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}={\mathrm{lim}}_{x\to 0}\frac{\frac{d}{dx}\left(\mathrm{sin}\left(x\right)\right)}{\frac{d}{dx}\left(x\right)}={\mathrm{lim}}_{x\to 0}\frac{\mathrm{cos}\left(x\right)}{1}=1$
2. Special Limits:

• Some limits have predefined values, like ${\mathrm{lim}}_{x\to 0}\frac{\mathrm{sin}\left(x\right)}{x}=1$ or ${\mathrm{lim}}_{x\to \mathrm{\infty }}\frac{1}{x}=0$. Memorizing these can expedite limit evaluations.

Indeterminate Forms:

• Expressions that do not immediately yield a determinate value (e.g., $\frac{0}{0}$, $\frac{\mathrm{\infty }}{\mathrm{\infty }}$, $0\cdot \mathrm{\infty }$) require further analysis using the methods mentioned.