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Limits Review, Exercises of Mathematics

Mathematics

A review of the concept of limits in calculus. It covers key topics such as graphical and analytical approaches to finding limits, special trigonometric limits, asymptotes, and continuity. The review provides examples and practice problems to help students solidify their understanding of limits. It serves as a comprehensive resource for students to prepare for exams or reinforce their knowledge of this fundamental calculus concept. Structured in a clear and organized manner, making it a valuable study aid for university-level calculus courses.

Typology: Exercises

2020/2021

Uploaded on 09/17/2023

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Name:_______________________________Date:_______________Period:_____
1Review–Limits
ReviewsdoNOTcoverallmaterialfromthelessonsbutwillhopefullyremindyouofkeypoints.Tobeprepared,
youmuststudyallpacketsfromUnit1.
1.1LimitsGraphically:
Whatisalimit?
The‐valueafunctionapproachesatagiven‐value.
Givethevalueofeachstatement.Ifthevaluedoesnotexist,write“doesnotexist”or“undefined.”
1.lim
→
󰇛󰇜
2.lim
→ 󰇛󰇜
3.󰇛3󰇜
4.󰇛2󰇜
5.lim
→
󰇛󰇜
6. lim
→
󰇛󰇜
7.󰇛1󰇜
8. lim
→
󰇛󰇜
1.2LimitsAnalytically:
Findingalimit:
1. DirectSubstitution.
2. Simplifyandthentrydirectsubstitution.
a. FactorandCancel.
b. Rationalizeifyouseesquareroots.
3. L’Hôpital’sRule(forindeterminateforms
or
)
SpecialTrigLimits:
lim
→
sin

or lim
→
sin
lim
→
1cos
or lim
→
cos 1
Evaluateeachlimit.
9. lim
→󰇛232
󰇜
10.lim
→ √7 42
11. lim
→ 2
12. lim
→


    



x
y
Review
pf2

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Download Limits Review and more Exercises Mathematics in PDF only on Docsity!

Name: _______________________________ Date:_______________ Period: _____

1 Review – Limits

Reviews do NOT cover all material from the lessons but will hopefully remind you of key points. To be prepared, you must study all packets from Unit 1.

1.1 Limits Graphically:

What is a limit? The ࢟ ‐value a function approaches at a given ݔ‐value.

Give the value of each statement. If the value does not exist, write “ does not exist ” or “ undefined .”

  1. (^) ݂௫→ଷlim ሺݔሻ ൌ
  2. (^) ݂௫→ଵlim ሺݔሻ ൌ
  3. ݂ ሺ3ሻ ൌ
  4. ݂ ሺെ2ሻ ൌ
    1. lim݂௫→ଶ ሺݔሻ ൌ
    2. (^) ௫→ିଶlim ݂శ ሺݔሻ ൌ
    3. ݂ ሺ1ሻ ൌ
    4. (^) ௫→ିଶlim ݂ష ሺݔሻ ൌ

1.2 Limits Analytically:

Finding a limit:

  1. Direct Substitution.
  2. Simplify and then try direct substitution. a. Factor and Cancel. b. Rationalize if you see square roots.
  3. L’Hôpital’s Rule (for indeterminate forms ଴଴ or ஶஶ )

Special Trig Limits:

௫→଴^ lim

sin ݔ ݔ

ൌ (^) or lim௫→଴

sin ݔ

lim ௫→଴

1 െ cos ݔ ݔ

ൌ or^ lim ௫→଴

cos ݔെ 1 ݔ

Evaluate each limit.

  1. (^) ௫→ିସlim ݔሺ2 ଶ^ ൅ 3 ݔെ 2ሻ (^) 10. lim ௫→ଵ √7 ݔ൅ 42^ 11. (^) ௫→ଵଷlim 2 12. lim ௫→ଵ଴

௫ మି^ ହ௫ିହ଴ ௫ିଵ଴

         







x

y

Review

  1. lim௫→଴√௫ାଵଽି௫^ √ଵଽ 14. lim ௫→଴

భೣ శభି ଵ ௫

  1. lim ௫→଴

ୱ୧୬ሺ଻௫ሻ ଵଵ௫ 16.^ lim௫→଴

ୱ୧୬మ^ ሺଷ௫ሻ ୱ୧୬మ^ ሺହ௫ሻ

1.3 Asymptotes:

Vertical Asymptotes: If the denominator equals 0 , then there is a hole or a vertical asymptote. If the factor does not cancel, then it’s a vertical asymptote.

One‐sided limits at vertical asymptotes approach െ∞ or ∞.

Horizontal asymptotes:

௫→ஶ^ lim௙ሺ௫ሻ௚ሺ௫ሻ^ will produce a horizontal asymptote at  ݕൌ 0 if ݃ increases faster than ݂.  ൌ ݕ ௔௕ if ݃ and ݂ are increasing at the relative same amount where ܽ and ܾ are the coefficients of the fastest growing terms.

Don’t forget to check the left and right sides when looking for horizontal asymptotes.

Evaluate each limit. Find all horizontal asymptotes.

  1. (^) ௫→ஶlim ସ௫^

ఱି (^) ଶ௫ మ (^) ାଷ ଷ௫ మ^ ାଶ௫ ఱି^ ௫ ర

  1. (^) ௫→ஶlim ݔ ହ^ 3ି ௫ 19. lim ௫→ஶ sin^

௫ାଷగ௫ మ ଶ௫ మ^ 20.^ ݂ ሺݔሻ ൌ^

√ଵ଺௫ ల^ ା௫ య^ ାହ௫ ହ௫ యି଼^ ௫

1.4 Continuity:

Types of Discontinuities:

  1. Removable (hole).
  2. Discontinuity due to vertical asymptote.
  3. Jump discontinuity.

Finding Domain:

Restrictions occur with two scenarios:

  1. Denominators can’t be zero.
  2. Even radicals can’t be negative.

Don’t forget the Intermediate Value Theorem (for continuous functions)! What is it and what does it tell us?