Algebraic limits

Evaluation of Algebraic Limits:

Objective: Algebraic limits involve determining the value that a function approaches as the independent variable approaches a specific value or infinity.

Methods for Evaluating Algebraic Limits:

1. Direct Substitution:

• This method applies when substituting the value of $x$into the function does not result in an undefined expression or division by zero.
• Example: ${\mathrm{lim}}_{x\to 2}\left(3x-1\right)=3\left(2\right)-1=5$
2. Factorization and Simplification:

• Useful for functions involving polynomials or rational expressions, factorizing and simplifying can simplify the expression to evaluate the limit.
• Example: ${\mathrm{lim}}_{x\to 4}\frac{{x}^{2}-16}{x-4}={\mathrm{lim}}_{x\to 4}\frac{\left(x+4\right)\left(x-4\right)}{x-4}={\mathrm{lim}}_{x\to 4}\left(x+4\right)=8$
3. Conjugate Method:

• Applicable for limits involving radicals or complex conjugates; multiplying by the conjugate of the denominator can help eliminate radicals.
• Example: ${\mathrm{lim}}_{x\to 1}\frac{\sqrt{x}-1}{x-1}={\mathrm{lim}}_{x\to 1}\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{x-1\left(\sqrt{x}+1\right)}={\mathrm{lim}}_{x\to 1}\frac{x-1}{x-1\left(\sqrt{x}+1\right)}=\frac{1}{2}$
• Rationalization:

• Applicable when limits contain square roots or cube roots in the denominator, multiplying by the conjugate can help eliminate radicals.
• Example: ${\mathrm{lim}}_{x\to 0}\frac{\sqrt{1+x}-1}{x}={\mathrm{lim}}_{x\to 0}\frac{\left(\sqrt{1+x}-1\right)\left(\sqrt{1+x}+1\right)}{x\left(\sqrt{1+x}+1\right)}={\mathrm{lim}}_{x\to 0}\frac{x}{x\left(\sqrt{1+x}+1\right)}=\frac{1}{2}$
• Factoring and Cancelation of Terms:

• Identifying common factors in the numerator and denominator and canceling them can simplify expressions to evaluate limits.
• Example: ${\mathrm{lim}}_{x\to 3}\frac{{x}^{2}-9}{x-3}={\mathrm{lim}}_{x\to 3}\frac{\left(x+3\right)\left(x-3\right)}{x-3}={\mathrm{lim}}_{x\to 3}\left(x+3\right)=6$