# One-Sided Limits

## Key Questions

• A left-hand limit means the limit of a function as it approaches from the left-hand side.

On the other hand, A right-hand limit means the limit of a function as it approaches from the right-hand side.

When getting the limit of a function as it approaches a number, the idea is to check the behavior of the function as it approaches the number. We substitute values as close as possible to the number being approached.

The closest number is the number being approached itself. Hence, one usually just substitutes the number being approached to get the limit.

However, we cannot do this if the resulting value is undefined.
But we can still check its behavior as it approaches from one side.

One good example is ${\lim}_{x \to 0} \frac{1}{x}$.

When we substitute $x = 0$ into the function, the resulting value is undefined.

Let's check its limit as it approaches from the left-hand side

$f \left(x\right) = \frac{1}{x}$

$f \left(- 1\right) = \frac{1}{-} 1 = - 1$
$f \left(- \frac{1}{2}\right) = \frac{1}{- \frac{1}{2}} = - 2$
$f \left(- \frac{1}{10}\right) = \frac{1}{- \frac{1}{10}} = - 10$
$f \left(- \frac{1}{1000}\right) = \frac{1}{- \frac{1}{1000}} = - 1000$
$f \left(- \frac{1}{1000000}\right) = \frac{1}{- \frac{1}{1000000}} = - 1000000$

Notice that as we get closer and closer to $x = 0$ from the left-hand side, the resulting value we gets larger and larger (though negative). We can conclude that the limit as $x \to 0$ from the left-hand side is $- \infty$

Now let's check the limit from the right-hand side

$f \left(x\right) = \frac{1}{x}$

$f \left(1\right) = \frac{1}{1} = 1$
$f \left(\frac{1}{2}\right) = \frac{1}{\frac{1}{2}} = 2$
$f \left(\frac{1}{10}\right) = \frac{1}{\frac{1}{10}} = 10$
$f \left(\frac{1}{1000}\right) = \frac{1}{\frac{1}{1000}} = 1000$
$f \left(\frac{1}{1000000}\right) = \frac{1}{\frac{1}{1000000}} = 1000000$

The limit as $x \to 0$ from the right-hand side is $\infty$

When the left-hand side limit of a function is different from the right-hand side limit, we can conclude that the function is discontinuous at the number being approached.

• A left-hand limit means the limit of a function as it approaches from the left-hand side.

On the other hand, A right-hand limit means the limit of a function as it approaches from the right-hand side.

When getting the limit of a function as it approaches a number, the idea is to check the behavior of the function as it approaches the number. We substitute values as close as possible to the number being approached.

The closest number is the number being approached itself. Hence, one usually just substitutes the number being approached to get the limit.

However, we cannot do this if the resulting value is undefined.
But we can still check its behavior as it approaches from one side.

One good example is ${\lim}_{x \to 0} \frac{1}{x}$.

When we substitute $x = 0$ into the function, the resulting value is undefined.

Let's check its limit as it approaches from the left-hand side

$f \left(x\right) = \frac{1}{x}$

$f \left(- 1\right) = \frac{1}{-} 1 = - 1$
$f \left(- \frac{1}{2}\right) = \frac{1}{- \frac{1}{2}} = - 2$
$f \left(- \frac{1}{10}\right) = \frac{1}{- \frac{1}{10}} = - 10$
$f \left(- \frac{1}{1000}\right) = \frac{1}{- \frac{1}{1000}} = - 1000$
$f \left(- \frac{1}{1000000}\right) = \frac{1}{- \frac{1}{1000000}} = - 1000000$

Notice that as we get closer and closer to $x = 0$ from the left-hand side, the resulting value we gets larger and larger (though negative). We can conclude that the limit as $x \to 0$ from the left-hand side is $- \infty$

Now let's check the limit from the right-hand side

$f \left(x\right) = \frac{1}{x}$

$f \left(1\right) = \frac{1}{1} = 1$
$f \left(\frac{1}{2}\right) = \frac{1}{\frac{1}{2}} = 2$
$f \left(\frac{1}{10}\right) = \frac{1}{\frac{1}{10}} = 10$
$f \left(\frac{1}{1000}\right) = \frac{1}{\frac{1}{1000}} = 1000$
$f \left(\frac{1}{1000000}\right) = \frac{1}{\frac{1}{1000000}} = 1000000$

The limit as $x \to 0$ from the right-hand side is $\infty$

When the left-hand side limit of a function is different from the right-hand side limit, we can conclude that the function is discontinuous at the number being approached.

Observe the y value that the graph approaches.

#### Explanation:

In order to find a one-sided limit graphically, you much observe the graph to find the point at which the graph is approaching. As opposed to standard limits, one-sided limits are only dependent on one side of a given x value. Locate this x value on the graph and see where the graph is approaching from the side you want.

For example, look at the following graph f(x). graph{sqrt(x-5)+10 [-28.86, 28.88, -14.44, 14.42]}
On the graph, the right sided limit of f(5) is 10 because that is the y value being approached on the left side, but there is no left sided limit because there is no graph on that side.

On piecewise functions, this may not be the cases. It is not necessary for the left and right side limits to be the same number, or for f(c) to be defined.

• This is when you attempt to evaluate the limit of a function from either the left side or the right side.