Trigonometric limits

Evaluation of Trigonometric Limits:

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

Methods for Evaluating Trigonometric Limits:

  1. Direct Substitution:

    • For simple trigonometric expressions, direct substitution by substituting the value of x into the function can help evaluate the limit.
    • Example: limx0sin(x)x=sin(0)0=1 (using the fact that limx0sin(x)x=1
  2. Trigonometric Identities:

    • Using trigonometric identities like sin2(x)+cos2(x)=1 or tan(x)=sin(x)cos(x) can help simplify expressions.
    • Example: limxπ2cos(x)sin(x)=limxπ2cos(x)1cos2(x)=00(which is an indeterminate form)
  3. L'Hôpital's Rule:

    • When trigonometric limits result in indeterminate forms (00 or ), applying L'Hôpital's Rule by finding derivatives of the numerator and denominator can be useful.
    • Example: limx0tan(x)x=limx0ddx(tan(x))ddx(x)=limx0sec2(x)1=1
  4. Special Trigonometric Limits:

    • Memorizing special limits such as limx0sin(x)x=1 or limx0tan(x)x=1 can aid in evaluating limits faster.
  5. Trigonometric Substitutions:

    • In more complex expressions, substitution techniques involving trigonometric identities may be employed to simplify and evaluate limits.
    • Example: limxπ4sin(x)cos(x)sin(x)+cos(x)=limxπ42sin(xπ4)2cos(xπ4)=1