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Hamza

March 19, 2021

Chapters

Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Solution of exercise 1
Solution of exercise 2
Solution of exercise 3
Solution of exercise 4
Solution of exercise 5
Solution of exercise 6

Exercise 1

Prove that the sequence ${ a }_{ n } = \frac { 2n + 4 }{ n }$ has a limit of $2$ . Also, calculate the terms whose distance from $2$ is less than $0.1$ .

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Exercise 2

Prove that the sequence ${ a }_{ n } = \frac { 4n + 1 }{ n }$ has a limit of $4$ and calculate how many terms of the succession are not within $(4 - 0.001, 4 + 0.001)$ .

Exercise 3

Prove that the sequence ${ a }_{ n } = \frac { { n }^{ 2 } }{ { n}^{ 2 } + 3 }$ has a limit of $1$ and calculate how many terms of the succession are not within $(1 - 0.001, 1 + 0.001)$ .

Exercise 4

Prove that $\lim_{ \frac { 3n - 8 }{ 4n + 1 } } = \frac { 3 }{ 4 }$ . Also, calculate the terms whose distance from the limit is less than $0.01$ .

Exercise 5

Prove that the sequence ${ a }_{ n } = \frac { { n }^{ 2 } + 1 }{ 4 }$ has a limit of $+ \infty$ and determine how many terms in the sequence are less than a million?

Exercise 6

Prove that the sequence ${ a }_{ n } = - { n }^{ 2 }$ a has a limit of $- \infty$ . Also, what term of the sequence produces values of less than $-10,000$ ?

Solution of exercise 1

Prove that the sequence ${ a }_{ n } = \frac { 2n + 4 }{ n }$ has a limit of $2$ . Also, calculate the terms whose distance from $2$ is less than $0.1$ .

$\left | \frac { 2n + 4 }{ n } - 2 \right | < \frac { 1 }{ 10 }$

$\left | \frac { 2n + 4 - 2n }{ n } \right | < \frac { 1 }{ 10 }$

$\left | \frac { 4 }{ n } \right | < \frac { 1 }{ 10 }$

$\frac { 4 }{ n } < \frac { 1 }{ 10 }$

$n > 40$

From ${ a }_{ 41 }$ the distance to $2$ is less than $0.1$

Solution of exercise 2

Prove that the sequence ${ a }_{ n } = \frac { 4n + 1 }{ n }$ has a limit of $4$ and calculate how many terms of the succession are not within $(4 - 0.001, 4 + 0.001)$ .

$\left | \frac { 4n + 1 }{ n } - 4 \right | < \frac { 1 }{ 1000 }$

$\left | \frac { 4n + 1 -4n }{ n } \right | < \frac { 1 }{ 1000 }$

$\left | \frac { 1 }{ n } \right | < \frac { 1 }{ 1000 }$

$\frac { 1 }{ n } < \frac { 1 }{ 1000 }$

$n > 1000$

The first thousand terms of the sequence are out.

Solution of exercise 3

Prove that the sequence ${ a }_{ n } = \frac { { n }^{ 2 } }{ { n}^{ 2 } + 3 }$ has a limit of $1$ and calculate how many terms of the succession are not within $(1 - 0.001, 1 + 0.001)$ .

$\left | \frac { { n }^{ 2 } }{ { n }^{ 2 } + 3 } - 1 \right | < \frac { 1 }{ 1000 }$

$\left | \frac { -3 }{ { n }^{ 2 } + 3 } \right | < \frac { 1 }{ 1000 }$

$\frac { 3 }{ { n }^{ 2 } + 3 } < \frac { 1 }{ 1000 }$

${ n }^{ 2 } + 3 > 3000$

${ n }^{ 2 } > 2997$

$n > 54$

The first $54$ terms are out.

Solution of exercise 4

Prove that $\lim_{ \frac { 3n - 8 }{ 4n + 1 } } = \frac { 3 }{ 4 }$ . Also, calculate the terms whose distance from the limit is less than $0.01$ .

$\left | \frac { 3n - 8 }{ 4n + 1 } - \frac { 3 }{ 4 } \right | < \frac { 1 }{ 100 }$

$\left | \frac { 12n - 32 - 12n - 3 }{ 16n + 4 } \right | < \frac { 1 }{ 100 }$

$\left | \frac { -35 }{ 16n + 4 } \right | < \frac { 1 }{ 100 }$

$\frac { 35 }{ 16n + 4 }| < \frac { 1 }{ 100 }$

$16n + 4 > 3500$

$n > 218.5$

From ${ n }_{ 219 }$ the distance to the limit is less than $0.01$ .

Solution of exercise 5

Prove that the sequence ${ a }_{ n } = \frac { { n }^{ 2 } + 1 }{ 4 }$ has a limit of $+ \infty$ and determine how many terms in the sequence are less than a million?

$\frac { { n }^{ 2 } + 1 }{ 4 } > 1,000,000$

$n > \sqrt { 3,999,999 } \qquad n > 1,999$

The $1,999$ first terms of the sequence.

Solution of exercise 6

Prove that the sequence ${ a }_{ n } = - { n }^{ 2 }$ a has a limit of $- \infty$ . Also, what term of the sequence produces values of less than $-10,000$ ?

$-1, -4, -9, -16, -25, -36, -49, ...$

$- { n }^{ 2 } < - N \qquad n > \sqrt { N }$

If $N = 10,000$ , its square root is $100$ , therefore, ${ a }_{ 101 }$ a₁₀₁ will be less than $-10,000$ .

${ a }_{ 101 } = - { 101 }^{ 2 } = - 10,201$

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.

Comments

Limit of Sequence Problems

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Solution of exercise 4

Solution of exercise 5

Solution of exercise 6

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Theory

Exponent Rules

Absolute Value Function

Asymptotes

Bounded Functions

Bounded Sequence

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Coordinates in the Plane

Even and Odd Functions

Exponential Function

Graphing polynomial function I

Graphing logarithmic function

Graphing polynomial function II

Graphing rational function II

Graphing rational function IV

Graphing rational function V

Graphing rational function 8

Graphing exponential function I

Geometric Sequence

Graphing Functions

x- and y-Intercepts

Inverse Function

L’Hôpital’s Rule

Limit of a Sequence

Logarithmic Functions

Absolute and Relative Maxima and Minima

Maxima and Minima

nth Term

Optimization

Properties of Limits

Quadratic Function

Rational Functions

Sequences

Types of Functions

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Concave and Convex Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Geometric Sequence Problems

Piecewise Functions

Trigonometric Functions

Arithmetic Sequence Problems

Linear Function

Domain of a Function

Maxima, Minima and Inflection Points

Graphing exponential function II

Linear Functions Worksheet

Monotone Sequence

Radical Functions

Function

Formulas

Sequence Formulas

Exercises

Absolute Value Functions Worksheet

Asymptotes Worksheet

Even and Odd Functions Worksheet

Inflection Points Worksheet