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Limits Exercises: Estimating and Evaluating Limits, Exercises of Mathematics

Union Institute & University Mathematics

A collection of exercises aimed at helping students understand the concept of limits. The exercises involve estimating and evaluating limits using tables of values and simplifying fractions. The limits covered include those of simple functions, radicals, and rational expressions.

What you will learn

What is the limit of sin(x^2) as x approaches infinity?
What is the limit of (x^2 + x)/x as x approaches infinity?
What is the limit of (x^2 + 4x + 6)/(3x^2 + 1) as x approaches infinity?

What is the limit of x/x as x approaches infinity?

Typology: Exercises

2021/2022

Uploaded on 02/11/2022

cristelle 🇺🇸

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Exercises: Limits

1–4 Use a table of values to guess the limit.

1. lim

x→∞

x1/x2. lim

x→∞

x−px2+x

3. lim

x→∞1+1

√xx

4. lim

x→∞

sin(x2)

5. Use a table of values to estimate the following limit:

lim

x→∞x

x+2x

Your answer must be correct to four decimal places.

6. Use a table of values to estimate the following limit:

lim

x→∞

x

p3x2+1

Your answer must be correct to four decimal places.

7–14 Identify the largest terms in the numerator and

denominator, and use your answers to evaluate the limit.

7. lim

x→∞

x

1+4x28. lim

x→∞

x3+2

x+1

9. lim

x→∞

6x+1

2x+510. lim

x→∞

x2

1−x2

11. lim

x→∞

x2+4x+6

3x2+112. lim

x→∞

1−4x3

x2+2x+1

13. lim

x→∞

x2+3

2x

14. lim

x→∞

x4+3x

x5+1

15–18 Use a table of values to guess the limit.

15. lim

x→0

√x+25 −5

x

16. lim

x→0

4x−1

8x−1

17. lim

x→4

√x−2

x−418. lim

x→1

x5−1

x−1

19–22 Evaluate the limit by simplifying the fraction.

19. lim

x→0

(x+5)2−25

x

20. lim

x→0

(x+1)3−1

x

21. lim

x→3

x2−9

x−322. lim

x→2

x2−7x+10

x−2

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Exercises: Limits

1–4 Use a table of values to guess the limit.

(^) xlim→∞ x^1 /x^ 2. (^) xlim→∞ x −

x^2 + x

(^) xlim→∞

x

)x

(^) xlim→∞ sin(x^2 )
Use a table of values to estimate the following limit:

lim x→∞

x x + 2

)x

Your answer must be correct to four decimal places.

Use a table of values to estimate the following limit:

lim x→∞

x √ 3 x^2 + 1 Your answer must be correct to four decimal places.

7–14 Identify the largest terms in the numerator and denominator, and use your answers to evaluate the limit.

(^) xlim→∞ x 1 + 4 x^2
(^) xlim→∞ x^3 + 2 x + 1
(^) xlim→∞ 6 x + 1 2 x + 5
(^) xlim→∞ x^2 1 − x^2
(^) xlim→∞

x^2 + 4 x + 6 3 x^2 + 1

(^) xlim→∞

1 − 4 x^3 x^2 + 2 x + 1

lim x→∞

x^2 + 3 2 x^

lim x→∞

x^4 + 3 x x^5 + 1

15–18 Use a table of values to guess the limit.

lim x→ 0

x + 25 − 5 x

lim x→ 0

4 x^ − 1 8 x^ − 1

lim x→ 4

x − 2 x − 4

lim x→ 1

x^5 − 1 x − 1

19–22 Evaluate the limit by simplifying the fraction.

lim x→ 0

(x + 5 )^2 − 25 x

lim x→ 0

(x + 1 )^3 − 1 x

lim x→ 3

x^2 − 9 x − 3

lim x→ 2

x^2 − 7 x + 10 x − 2

Limits Exercises: Estimating and Evaluating Limits, Exercises of Mathematics

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Exercises: Limits

Answers

1. 1 2. − 0. 5 3. infinity 4. no limit 5. 0. 1353 6. 0. 5774

7. x/ 4 x^2 ; the limit is 0 8. x^3 /x; the limit is ∞ 9. 6 x/ 2 x; the limit is 3 10. x^2 / − x^2 ; the limit is − 1

11. x^2 / 3 x^2 ; the limit is 1/ 3 12. − 4 x^3 /x^2 ; the limit is −∞ 13. x^2 / 2 x; the limit is 0 14. 3 x/x^5 ; the limit is ∞