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Limits involving Trigonometric Functions. (from section 3.3). In the following examples we use the following two formulas (which you can use in exams.
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In the following examples we use the following two formulas (which you can use in exams freely):
Important Note: When calculating the limits involving trigonometric functions, always look for an expression like sin x^ xor (^) sinx x if x → 0 because in that case both of these have limit equal to 1.
Example (section 3.3 exercise 50): Evaluate lim x→ (^1) x^ sin( (^2) +x x^ − −^ 1) 2
Solution: This limit is of the form (^00) = lim x→ (^1) (x^ sin( − 1)(x^ −x + 2)1) = lim x→ 1 sin( (x x−^ − 1)^ 1) (x + 2)^1 =^ [ x lim→ 1 sin( (xx −^ − 1)^ 1)^ ] [ x lim→ (^1) (x + 2)^1 ] =^ [ θ lim→ 0 sin θ^ θ^ ] [ x lim→ 1 (x + 2)^1 ] change of variable θ = x − 1 = (1)(^13 ) =^13
Example : Evaluate lim x→ 5 2 tan( x (^2) − 6 xx^ − + 5^ 5)
Solution: This is of the form (^00) = lim x→ (^5) (^ 2 tan(x − 5)(xx^ − −^ 5) 1) = lim x→ 5 (x 2 −^ cos(sin( 5)(xxx−− 5)5)− 1) = lim x→ 5 (x − 5)(^ 2 sin(x − x1) cos(^ −^ 5)x − 5) = lim x→ 5 2 sin( (xx −^ − 5)^ 5) (x − 1) cos(^1 x − 5) = 2^ [ x lim→ 5 sin( (xx −^ − 5)^ 5)^ ] [ x lim→ (^5) (x − 1) cos(^1 x − 5)^ ] = 2^ [ θ lim→ 0 sin θ^ θ^ ] [ x lim→ (^5) (x − 1) cos(^1 x − 5)^ ] change of variable θ = x − 5 = 2(1) (^) (4) cos(0)^1 = 2(1) (^) (4)(1)^1 =^12
