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: limx3(2x1)=2(3)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: limx2x24x2=limx2(x+2)(x2)x2=limx2(x+2)=4
  3. Conjugate Method:

    • Useful for limits involving square roots or complex conjugates; multiplying by the conjugate of the denominator helps eliminate radicals.
    • Example: limx0x+93x=limx0(x+93)(x+9+3)x(x+9+3)=limx0xx(x+9+3)=16
  1. L'Hôpital's Rule:

    • Used for limits involving indeterminate forms such as 00 or . It applies to the ratio of derivatives of two functions.
    • Example: limx0sin(x)x=limx0ddx(sin(x))ddx(x)=limx0cos(x)1=1
  2. Special Limits:

    • Some limits have predefined values, like limx0sin(x)x=1 or limx1x=0. Memorizing these can expedite limit evaluations.

Indeterminate Forms:

  • Expressions that do not immediately yield a determinate value (e.g., 00, , 0) require further analysis using the methods mentioned.