Understanding Limits at Infinity and Horizontal Asymptotes in
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MATH AP
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Mathematics
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Dec 11, 2024
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4
Uploaded by PresidentDiscovery11862
Horizontal Asymptotes:(End-behavior)What does the ࠵?-value approach as the ࠵?-value approaches negative infinity AND positive infinity? Does it approach a specific number, or is it growing without bound? Basic Rules for Horizontal Asymptotes:DENOMINATORgrows faster means ୬୭୲ ୟୱ ୠ୧ୱ୳୮ୣ୰ ୢ୳୮ୣ୰ ୍ୋ ୬୳୫ୠୣ୰!=࠵?If the numerator and denominator grow EQUALLYfast, then you have ୍ୋ ୬୳୫ୠୣ୰!୍ୋ ୬୳୫ୠୣ୰! =࠵?If the NUMERATORgrows faster than the denominator, then you have ୍ୋ ୬୳୫ୠୣ୰!୬୭୲ ୟୱ ୠ୧ =∞First, you need to recognize which functions grow faster as ࠵?-values get larger and larger. Rank Fastest to Slowest ࠵?(࠵?)࠵? = 1࠵? = 10࠵? = 100࠵? = 10006࠵?ଶ110010,000105࠵?ଷ11,0001010ଽ4࠵?ଵ110ଵ10ଶ10ଷ32௫21,0241.26 × 10ଷ1.07 × 10ଷଵ2࠵?௫2.71822,0262.69 × 10ସଷREALLY BIG 14௫41.05 × 101.6 × 10SUPER-DUPER BIG 7ln ࠵?02.3034.6056.908Find the horizontal asymptote(s) of each function.4. ࠵? =௫మାସଷ௫ିହ5. ࠵? =௫ାସଷ௫ିହ6. ࠵? =௫ାସଷ௫మିହ7. ࠵?(࠵?) =(௫ାହ)(௫ିଶ)(ସ௫ିଷ)మ8. ࠵? =√ସ௫మା௫ିଶଷ௫ିଵ9. ࠵? =√ସ௫రା௫ିଶଷ௫మିଵEvaluate the limit.10. lim௫→ஶ−4࠵?భೣ11. lim௫→ஶ5࠵?ି௫1.15 Limits at Infinity and Horizontal AsymptotesWrite your questions and thoughts here! Calculus Notes
Trig Functions’ Horizontal Asymptotes:Evaluate the limit.12. lim௫→ିஶୱ୧୬ ௫௫13. lim௫→ஶ−3 cosଵ௫14. lim௫→ஶsin ࠵?15. lim௫→ஶ5࠵? cos ࠵?1.15 Limits at Infinity and Horizontal AsymptotesCalculus Identify all horizontal asymptotes of each function. 1. ࠵?(࠵?) =(ଶ௫ିସ)(ଷ௫ାଵ)(ଶ௫ି)మ2. ࠵?(࠵?) =(ଶ௫ାହ)(ଶି௫)(ଷ௫ିଶ)మ3. ࠵?(࠵?) =(ହ௫ିଵ)(௫మାସ)(ହ௫ାଵ)మ4. ࠵?(࠵?) =√ଶହ௫రାଶ௫௫మ5. ࠵?(࠵?) =√௫లାଷ௫మା௫௫యାସ௫మ6. ࠵?(࠵?) =√ଽ௫ఴିଶ௫యି௫ଶ௫రିଵ௫+ 37. ࠵?(࠵?) =ଷ௫√ଽ௫మିଵEvaluate each limit. 8. lim௫→ஶି௫ାଶ௫మାଶ௫ାଶ9. lim௫→ஶቀsinଵ௫−௫మାଶ௫ଷ௫మቁ10. lim௫→ஶቀ5 cosଵ௫ቁ11. lim௫→ஶ௫ళସೣ− 512. lim௫→ஶ3ି௫+ 213. lim௫→ஶ−3࠵? cos ࠵?14. lim௫→ஶ2࠵? sin ࠵?15. lim௫→ஶଽ௫రାସ௫యାଷ௫ళାଶ௫రାଶ௫య16. lim௫→ିஶଷ௫మିହ௫ାଵଵ௫మିଶ௫17. lim௫→ஶcos ቀଶ௫ିగ௫మ௫మቁ18. lim௫→ஶቀୱ୧୬ ௫௫− 4ቁ19. lim௫→ஶି௫రିଷ௫మି଼ହ௫రା௫ାଵଷ20. lim௫→ஶ௫యି௫మା଼௫మା௫ିଶ21. lim௫→ஶ࠵?ଶ2ି௫22. lim௫→ஶళଽೣ23. lim௫→ିஶଷ௫మିହ௫ళା௫ళିଵହ௫రPractice Write your questions and thoughts here! 5x34x=y=3/225X2noHA*#y=5y=5y=-vy=0sin(0)-4/3=- 2Gros(0):505=-55x+2DNEONEO2ros1)=-0-4=-4*=0O
24. lim௫→ஶଶ௫రାଷ௫మାଵହ௫మା௫ିଵ25. lim௫→ஶቀୱ୧୬ ௫௫+ 2ቁ26. lim௫→ஶcos ቀ௫ఱೣቁ + 427. lim௫→ஶଷ௫లିହ௫యା௫యା௫ఴିଶ௫ర28. lim௫→ஶsin(2࠵?)29. lim௫→ஶcos ቆగ௫మା√మమ௫ହିଶ௫మቇ30. lim௫→ஶcos ቆ√మమ௫ିగ௫మ௫మି௫యାଶቇ31. Which of the following functions grows the fastest? (A) ࠵?(࠵?) = ቀଵଶቁ௨(B) ࠵?(࠵?) = ࠵?ଵ+ ࠵?ଽଽ(C) ࠵?(࠵?) = 4௨(D) ࠵?(࠵?) = 200࠵?௨(E) ࠵?(࠵?) = 3௨+ ࠵?ଷ32. Suppose that ࠵?(࠵?) = sinଶ࠵? √࠵?+ 4, and lim௫→ஶ(௫)(௫)= 0. Which of the following functions could be ࠵?? (A) ࠵?(B) ࠵?ଶ(C) ࠵?ଷ(D) ࠵?ସ(E) ln ࠵?33. Which of the following statements are true for the function ࠵?(࠵?) =ଶ௫యାଷ௫ାଵଶೣI. ࠵?(࠵?)has a horizontal asymptote of ࠵? = 1II. ࠵?(࠵?)has a horizontal asymptote of ࠵? = 0III. ࠵?(࠵?)has a vertical asymptote of ࠵? = 0(A) I only (B) II only (C) III only (D) I and III only (E) II and III only Test Prep 1.15 Limits at Infinity and Horizontal Asymptotes0+2=2cos(0)+4SOO&
34. Which of the following functions has both a vertical and horizontal asymptote? (A) ࠵?(࠵?) =ଵଵାషೣ(B) ࠵?(࠵?) = tan ࠵?(C) ࠵?(࠵?) =௫௫మାଶ(D) ࠵?(࠵?) =௫௫మିଶ(E) ࠵?(࠵?) =௫మାଶ௫35. The function ࠵?(࠵?) = ቐ௫మାଶ௫ାଷ௫మିଵ, ࠵? ≥ 0௫ೣ, ࠵? < 0has which of the following asymptotes? (A) ࠵? = 0only. (B) ࠵? = 1only. (C) ࠵? = 1, ࠵? = 1only. (D) ࠵? = 1, ࠵? = ±1only. (E) ࠵? = 0, ࠵? = 1, ࠵? = ±1. 36. If the function ࠵?(࠵?) =ି௫యା௫మା௫ାௗషೣି௪௫యା௪has a horizontal asymptote of ࠵? = 2and a vertical asymptote of ࠵? = 0, then ࠵? − ࠵? =(A) −1(B) 0(C) 1(D) ∞(E) The limit does not exist. 37. What are all horizontal asymptotes of the graph of ࠵? =ହାଶೣଵିଶೣin the ࠵?࠵?-plane? (A) ࠵? = −1only (B) ࠵? = 0only (C) ࠵? = 5only (D) ࠵? = −1and ࠵? = 0(E) ࠵? = −1and ࠵? = 5