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 xinto the function does not result in an undefined expression or division by zero.
    • Example: limx2(3x1)=3(2)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: limx4x216x4=limx4(x+4)(x4)x4=limx4(x+4)=8
  3. Conjugate Method:

    • Applicable for limits involving radicals or complex conjugates; multiplying by the conjugate of the denominator can help eliminate radicals.
    • Example: limx1x1x1=limx1(x1)(x+1)x1(x+1)=limx1x1x1(x+1)=12
  • Rationalization:

    • Applicable when limits contain square roots or cube roots in the denominator, multiplying by the conjugate can help eliminate radicals.
    • Example: limx01+x1x=limx0(1+x1)(1+x+1)x(1+x+1)=limx0xx(1+x+1)=12
  • Factoring and Cancelation of Terms:

    • Identifying common factors in the numerator and denominator and canceling them can simplify expressions to evaluate limits.
    • Example: limx3x29x3=limx3(x+3)(x3)x3=limx3(x+3)=6