Chapters

The concept of infinity comes from division by zero. Imagine you have 15 pencils, you need to divide them into zero people, how much did each person get? It would be nothing! Because nothing is given to them. In mathematics, when you divide any number by zero, it will result in infinity. For example, divide 15 by 0, the division will go on forever but you will achieve nothing. The answer will be undefined. Finding the limit of an expression that is divided by zero can be either + \infty, - \infty or no limit.

For example, you are asked to find the limit of \frac { x - 1 }{ x + 1 } which is approaching -1. Let's replace x with -1 and find what would we get:

\lim_{ x \rightarrow -1 } \frac { x - 1 }{ x + 1 } = \frac { - 1 - 1 }{ - 1 + 1 } = \frac { -2 }{ 0 }

One thing is clear, you will get infinity but what is the sign? Is it either positive infinity or negative infinity? To find that, we will be performing the side limits to determine the sign of \infty.

If x has a value that approaches -1 from the left, -1.1, both the numerator and denominator are negative and the left side limit is: + \infty.

\lim_{ x \rightarrow -1 } \frac { x - 1 }{ x + 1 } = \frac { (-) }{ (-) } = \infty

If x has a value that approaches -1 from the right, -0.9, the numerator will be negative, the denominator positive and therefore the right-side limit will be: - \infty.

\lim_{ x \rightarrow -1 } \frac { x - 1 }{ x + 1 } = \frac { (-) }{ (+) } = - \infty

The next step is to find whether the function has an output on that input. To best way to find that is to check the side limits, if they coincide, that means the function has an output for that input. Since the side limits do not coincide, the function has no limit as x = - 1

Example

\lim_{ x \rightarrow 0 } \frac { 1 }{ { x }^{ 2 } }

\lim_{ x \rightarrow 0 } \frac { 1 }{ { x }^{ 2 } } = \frac { 1 }{ 0 } = \infty

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

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

\lim_{ x \rightarrow 0 } \frac { 1 }{ { x }^{ 2 } } = \infty

 

\lim_{ x \rightarrow 0 } (- \frac { 1 }{ { x }^{ 2 } })

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

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

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

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

Need a Maths teacher?

Did you like the article?

1 Star2 Stars3 Stars4 Stars5 Stars 5.00/5 - 2 vote(s)
Loading...

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.