Learn Maths from the best

Chapters

Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Solution of exercise 1
Solution of exercise 2
Solution of exercise 3
Solution of exercise 4
Solution of exercise 5
Solution of exercise 6
Solution of exercise 7
Solution of exercise 8
Solution of exercise 9
Solution of exercise 10
Solution of exercise 11

Exercise 1

$f(x) = \frac { { x }^{ 2 } + 2 }{ x - 2 }$

The best Maths tutors available

Exercise 2

$f(x) = \frac { { x }^{ 3 } }{ { (x - 1) }^{ 2 } }$

Exercise 3

$f(x) = \frac { { x }^{ 4 } + 1 }{ { x }^{ 2 } }$

Exercise 4

$f(x) = \frac { { x }^{ 2 } }{ 2 - x }$

Exercise 5

$f(x) = \frac { x }{ 1 + { x }^{ 2 } }$

Exercise 6

$f(x) = \frac { { x }^{ 2 } - 3x + 2 }{ { x }^{ 2 } + 1 }$

Exercise 7

$f(x) = \frac { { x }^{ 2 } }{ \sqrt { { x }^{ 2 } - 1 } }$

Exercise 8

$f(x) = { e }^{ \frac { 1 }{ x } }$

Exercise 9

$f(x) = (x - 1){ e }^{ -x }$

Exercise 10

$f(x) = \frac { 1 }{ 2 \sqrt { 2 \pi }} { e }^{ - \frac { 1 }{ 2 } { x }^{ 2 } }$

Exercise 11

$f(x) = \frac { \ln { x } }{ x }$

Solution of exercise 1

$f(x) = \frac { { x }^{ 2 } + 2 }{ x - 2 }$

Horizontal asymptotes

$\lim_{ x \rightarrow \infty } \frac { { x }^{ 2 } + 2 }{ x - 2 } = \infty$

Vertical asymptotes.

$\lim_{ x \rightarrow 2 } \frac { { x }^{ 2 } + 2 }{ x - 2 } = \infty$

Oblique asymptotes.

$m = \lim_{ x \rightarrow \infty } \frac { \frac { { x }^{ 2 } + 2 }{ x - 2 } }{ x } = \lim_{ x \rightarrow \infty } \frac { { x }^{ 2 } + 2 }{ { x }^{ 2 } - 2x } = 1$

$n = \lim_{ x \rightarrow \infty } ( \frac { { x }^{ 2 } + 2 }{ x - 2 } - (1 \times x) ) = \lim_{ x \rightarrow \infty } \frac { 2x + 2 }{ x - 2 } = 2$

$y = x + 2$

Solution of exercise 2

$f(x) = \frac { { x }^{ 3 } }{ { (x - 1) }^{ 2 } }$

Horizontal asymptotes

$\lim_{ x \rightarrow \infty } \frac { { x }^{ 3 } }{ { (x - 1) }^{ 2 } } = \infty \qquad No$

Vertical asymptotes.

$\lim_{ x \rightarrow 1 } \frac { { x }^{ 3 } }{ { (x - 1) }^{ 2 } } = \infty \qquad x = 1$

Oblique asymptotes.

$m = \lim_{ x \rightarrow \infty } \frac { \frac { { x }^{ 3 } }{ { (x - 1) }^{ 2 } } }{ x } = 1$

$n = \lim_{ x \rightarrow \infty } [\frac { { x }^{ 3 } }{ { (x - 1) }^{ 2 } } - x] = 2$

$y = x + 2$

Solution of exercise 3

$f(x) = \frac { { x }^{ 4 } + 1 }{ { x }^{ 2 } }$

Horizontal asymptotes

$\lim_{ x \rightarrow \infty } f(x) = \frac { { x }^{ 4 } + 1 }{ { x }^{ 2 } } = \infty \qquad No$

Vertical asymptotes.

$\lim_{ x \rightarrow 0 } f(x) = \frac { { x }^{ 4 } + 1 }{ { x }^{ 2 } } = \infty \qquad x = 0$

Oblique asymptotes.

$m = \lim_{ x \rightarrow \infty } \frac { \frac { { x }^{ 4 } + 1 }{ { x }^{ 2 } } }{ x } = \lim_{ x \rightarrow \infty } \frac { { x }^{ 4 } + 1 }{ { x }^{ 3 } } = \infty \qquad No$

Solution of exercise 4

$f(x) = \frac { { x }^{ 2 } }{ 2 - x }$

Horizontal asymptotes

$\lim_{ x \rightarrow \infty } \frac { { x }^{ 2 } }{ 2 - x } = \infty \qquad No$

Vertical asymptotes.

$\lim_{ x \rightarrow 2 } \frac { { x }^{ 2 } }{ 2 - x } = \infty \qquad x = 2$

Oblique asymptotes.

$m = \lim_{ x \rightarrow \infty } \frac { \frac { { x }^{ 2 } }{ 2 - x } }{ x } = -1$

$n = \lim_{ x \rightarrow \infty } (\frac { { x }^{ 2 } }{ 2 - x } + x) = -2$

$y = -x -2$

Solution of exercise 5

$f(x) = \frac { x }{ 1 + { x }^{ 2 } }$

Horizontal asymptotes

$\lim_{ x \rightarrow \infty } \frac { x }{ 1 + { x }^{ 2 } } = 0 \qquad y = 0$

No vertical or oblique asymptote.

Solution of exercise 6

$f(x) = \frac { { x }^{ 2 } - 3x + 2 }{ { x }^{ 2 } + 1 }$

Horizontal asymptotes

$\lim_{ x \rightarrow \infty } \frac { { x }^{ 2 } - 3x + 2 }{ { x }^{ 2 } + 1 } = 1 \qquad y = 1$

No vertical or oblique asymptote.

Solution of exercise 7

$f(x) = \frac { { x }^{ 2 } }{ \sqrt { { x }^{ 2 } - 1 } }$

${ x }^{ 2 } - 1 } > 0 \qquad x = \pm 1 \qquad D = (- \infty, -1)U(1, \infty)$

Horizontal asymptotes.

$\lim_{x \rightarrow \infty } \frac { { x }^{ 2 } }{ \sqrt { { x }^{ 2 } - 1 } } = \infty$

No Horizontal Asymptote.

Vertical asymptotes.

$\lim_{x \rightarrow -1 } \frac { { x }^{ 2 } }{ \sqrt { { x }^{ 2 } - 1 } } = \infty \qquad x = -1$

$\lim_{x \rightarrow 1 } \frac { { x }^{ 2 } }{ \sqrt { { x }^{ 2 } - 1 } } = \infty \qquad x = 1$

Oblique asymptotes.

$m = \lim_{x \rightarrow \infty } \frac { \frac { { x }^{ 2 } }{ \sqrt { { x }^{ 2 } - 1 } } }{ x } = \lim_{x \rightarrow \infty } \frac { { x }^{ 2 } }{ x \sqrt { { x }^{ 2 } - 1 } } = \lim_{x \rightarrow \infty } \frac { { x }^{ 2 } }{ \sqrt { { x }^{ 4 } - { x }^{ 2 } } } = 1$

$n = \lim_{x \rightarrow \infty } (\frac { { x }^{ 2 } }{ \sqrt { { x }^{ 2 } - 1 } } - x) = \lim_{x \rightarrow \infty } \frac { { x }^{ 2 } - x \sqrt { { x }^{ 2 } - 1 } }{ \sqrt { { x }^{ 2 } - 1 } }$

$= \lim_{x \rightarrow \infty } \frac { { x }^{ 2 } - \sqrt { { x }^{ 4 } - { x }^{ 2 } } }{ \sqrt { { x }^{ 2 } - 1 } } = \lim_{x \rightarrow \infty } \frac { ({ x }^{ 2 } - \sqrt { { x }^{ 4 } - { x }^{ 2 } })( { x }^{ 2 } + \sqrt { { x }^{ 4 } - { x }^{ 2 } } ) }{ \sqrt { { x }^{ 2 } - 1 } ({ x }^{ 2 } + \sqrt { { x }^{ 4 } - { x }^{ 2 } }) }$

$\lim_{x \rightarrow \infty } \frac { { x }^{ 4 } - ({ x }^{ 4 } - { x }^{ 2 }) }{ \sqrt { { x }^{ 2 } - 1 } ({ x }^{ 2 } + \sqrt { { x }^{ 4 } - { x }^{ 2 } }) } = \lim_{x \rightarrow \infty } \frac { { x }^{ 2 } }{ \sqrt { { x }^{ 2 } - 1 } ({ x }^{ 2 } + \sqrt { { x }^{ 4 } - { x }^{ 2 } }) } = 0$

$y = x$

Solution of exercise 8

$f(x) = { e }^{ \frac { 1 }{ x } }$

Horizontal asymptotes

$\lim_{x \rightarrow \infty } { e }^{ \frac { 1 }{ x } } = { e }^{ 0 } = 1 \qquad y = 1$

Vertical asymptotes.

$\lim_{x \rightarrow 0 } { e }^{ \frac { 1 }{ x } } =\infty \qquad x = 0$

Solution of exercise 9

$f(x) = (x - 1){ e }^{ -x }$

Horizontal asymptotes

$\lim_{x \rightarrow \infty } (x - 1){ e }^{ -x } = \lim_{x \rightarrow \infty } \frac { x - 1 }{ { e }^{ -x } } = 0 \qquad y = 0$

No vertical or oblique asymptote.

Solution of exercise 10

$f(x) = \frac { 1 }{ 2 \sqrt { 2 \pi }} { e }^{ - \frac { 1 }{ 2 } { x }^{ 2 } }$

Horizontal asymptotes

$\lim_{x \rightarrow \infty } \frac { 1 }{ 2 \sqrt { 2 \pi }} { e }^{ - \frac { 1 }{ 2 } { x }^{ 2 } } = 0 \qquad y = 0$

No vertical or oblique asymptote.

Solution of exercise 11

$f(x) = \frac { \ln { x } }{ x }$

Horizontal asymptotes

$\lim_{x \rightarrow \infty } \frac { \ln { x } }{ x } = 0 \qquad y = 0$

Vertical asymptotes.

$\lim_{x \rightarrow { 0 }^{ + } } \frac { \ln { x } }{ x } = \infty \qquad x = 0$

The platform that connects tutors and students

Did you like this article? Rate it!

5.00 (2 rating(s))

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.

Asymptotes Worksheet

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Solution of exercise 4

Solution of exercise 5

Solution of exercise 6

Solution of exercise 7

Solution of exercise 8

Solution of exercise 9

Solution of exercise 10

Solution of exercise 11

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing rational function IV

Graphing 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

Geometric Sequence

x- and y-Intercepts

Bounded Sequence

Graphing rational function 8

L’Hôpital’s Rule

Graphing polynomial function I

Graphing exponential function I

Absolute and Relative Maxima and Minima

Maxima and Minima

Types of Functions

Asymptotes

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Inverse Function

Graphing rational function II

Graphing logarithmic function

Graphing polynomial function II

Quadratic Function

Absolute Value Function

Rational Functions

Sequences

Bounded Functions