Definition of Trigonometric functions Six trigonometric value 180 = π Radian) π 90 = 2 π π π cos π = π sin π = b: opposite θ a: adjacent π π π cot π = π tan π = 360 = 2π in Radian (sin π) + (cos π) = 1 (tan π) + 1 = (sec π) (cot π) + 1 = (csc π) π(in Radian) 0 =0 sec π = π π csc π = π π π = 30 = 1 45 = θ π π π cot π = π tan π = csc π = π π π = 60 = θ π π π cos π = π π π 60 = 360 2π π = = 6 6 3 45 = 360 2π π = = 8 8 4 30 = 360 2π π = = 12 12 6 0 =0 sin π = sec π = (in 5 θ 90 = 180 = π Section 2-1 Limit and Example Exercise Limit laws limπ₯ + 1 = 2 Definition: The limit of f → at a is L, is defined by lim2π₯ + 1 = → π₯ −1 = → π₯−1 lim lim π(π₯) = πΏ → lim → = lim π(π₯) = lim π(π₯) π₯ − 25 = π₯−5 → → (Two sided limit) One-sided limit: lim π(π₯): Right hand → limit lim π(π₯): left hand → limit where π is a constant Section 2-2 Limit lim π = π Theorems lim π₯ = π 1. Constant law 2. Identity law 3. Constant multiple lim ππ(π₯) = πlim π(π₯), constant multiple 4. Sum of limits lim (π(π₯) + π(π₯))= lim π(π₯) + lim π(π₯) 5. Difference of limits lim (π(π₯) − π(π₯))= lim π(π₯) − lim π(π₯) 6. Multiplication of lim(π(π₯)π(π₯))= lim π(π₯)lim π(π₯) → → limits 7. Division of limits → → → → → → → lim → → → ( ) ( ) = → → → → ( ) ( ) lim (π(π₯)) = (lim π(π₯)) 8. n-power 9. nth-root → → lim π(π₯) = lim π(π₯) ,lim π(π₯) ≥ 0 , if n is even → 1. Constant law lim π = π → → lim 5 = 5 lim 4 = lim π₯ = 2 lim π₯ = lim 7π₯ = 7(2) = 14 lim 6π₯ = → → → 2. Identity law lim π₯ = π → → → 3. Constant multiple lim π(π₯) = πΏ → → → Then lim ππ(π₯) = ππΏ → Suppose lim π(π₯) = πΏ and lim π(π₯) = π (πΏ and π exist as finite numbers. ) → 4. Sum lim π(π₯) = πΏ and → lim π₯ + 5 = 2 + 5 = 7 lim π₯ + 4 = → → → 6 lim π(π₯) = π → Then lim π(π₯) + π(π₯) = → πΏ+π 5. Subtract lim π(π₯) − π(π₯) = πΏ − π lim(π₯ − 5) = 2 − 5 = −3 lim π₯ − 4 = lim π₯ = 2 = 4 lim6π₯ = lim π₯ − 5 −3 = → π₯ 4 lim lim(π₯ − 5)3 = (2 − 5) = −27 lim(π₯ − 4)3 = lim lim 6π₯ + 10 = → → → 6. Multiplication lim π(π₯)π(π₯) = πΏπ → → → 7. Division lim → ( ) ( ) = ,if π ≠ 0 8. Power lim (π(π₯)) = πΏ π₯−4 = → 6π₯ → → → 9. Radical −16π₯ = lim → (−16 ∗ 4) = −4 → → lim π(π₯) = lim (π(π₯)) → → = √πΏ lim 2π₯ + 1 = lim √9 = 3 → lim 3π₯ + 4 = → → L>0 if n is even 10. indeterminate form ( ) |π₯| = → π₯ lim |π₯ − 1| π₯ = lim = 1, π₯ >lim → π₯−1 → π₯ −π₯ lim = −1, π₯ → π₯ π₯−1 lim = 1, π₯ > 1 → π₯−1 −(π₯ − 1) lim = −1, π₯ < 1 → π₯−1 |1 − π₯| = → π₯−1 lim lim |1 − π₯| = π₯−1 lim |1 − π₯| = π₯−1 → → 11. Factorization: indeterminate form ( ) lim → (π₯ + 2)(π₯ − 2π₯ + 4) π₯ +8 = lim → π₯+2 π₯+2 → → = lim (π₯ − 2π₯ + 4) = 4 − 2(−2) + 4 π₯ −4 → π₯−2 → lim = lim lim lim = 12 → = π₯ −9 π₯ + 2π₯ − 15 (π₯ − 2)(π₯ + 2) π₯−2 = limπ₯ + 2 = 4 → 12. Rationalize: indeterminate form ( ) √π₯ + 9 − 3 → π₯ lim lim → (√π₯ + 9 − 3)(√π₯ + 9 + 3) √π₯ + 9 − 3 = lim → π₯ π₯ (√π₯ + 9 + 3) = lim → = lim → (π₯ + 9) − 9 π₯ (√π₯ + 9 + 3) 1 √π₯ + 9 + 3 7 = 1 6 √π₯ + 4 − 2 → π₯ lim Rationalization lim → Section 2.4 √9 + π₯ − 3 π₯ lim → sin π₯ =1 → π₯ sin 2π₯ = → π₯ lim Trigonometric Limits √5 + π₯ − √5 π₯ lim sin 2π₯ = → sin 3π₯ lim lim → cos π₯ − 1 =0 π₯ lim = → indeterminate form ( ) lim sin2 π = 3π lim sin5 π = sin3 π lim tan5 π = sin3 π lim tan5 π = π → sin π =1 → π lim -0.1 θ sin π 0.9983 π -0.01 -0.001 0 0.001 0.01 0.1 0.99998 0.99999 X 0.99999 0.99998 0.99833 → → → indeterminate form ( ) lim → (cos π − 1)(cos π + 1) cos π − 1 = lim → → π π(cos π + 1) lim cos π − 1 =0 π −(sin π) → π(cos π + 1) = lim (− sin π) sin π =0 → π(cos π + 1) = lim lim π(π₯) = πΏ One sided limit → lim π(π₯) = πΏ → Section 2.5 lim π(π₯) = ±∞ Limit involving lim π(π₯) = ±∞ Infinity lim π(π₯) = ±∞ → → lim π(π₯) = ±∞ → x 1 π₯ -0.0001 0 0.0001 0.01 0.1 -10 -100 -10000 X 10000 100 10 = lim 1 = π₯−1 lim 1 = π₯−1 lim 1 = π₯−1 lim 1 = π₯−1 → → lim = 0, → does not exit -0.01 =∞ → lim → lim → = −∞ =0 : : 1 π₯−π lim → -0.1 lim lim → Limit involving infinity → π₯−2 π₯ −4 → → lim lim x=0: is a vertical asymptote → y=0: is a horizontal asymptote → 8