calculuspowerup.com

Limits on the Endpoints or Outside the Domain

April 8th, 2009 |

Author: Merrin

Certain trick problems may ask you to evaluate limits at the endpoint of a function or outside of the functions domain. In the case of an endpoint, one of the one sided limits will not exist. If the approach to a point lies outside the domain of a function similarly the limit does not exist. Both of these cases are different from examining a limit at a singularity or discontinuity.

Example 1. Find the limit

$\mathop{\lim }\limits_{x\to 0} \ln x$

Solution 1. We can take the right hand limit of ln x and find that it is negative infinity. We can’t take the left hand limit because ln x is only defined for x > 0. As a side note, ln (0) is also not a number, but this does not factor in to our calculation of the limit.

$\mathop{\lim }\limits_{x\to 0^{-} } \ln x=DNE\quad \quad \quad \mathop{\lim }\limits_{x\to 0^{+} } \ln x=-\infty$

So

$\mathop{\lim }\limits_{x\to 0} \ln x=DNE$

Example 2. Define a function as

$f(x)=x \quad x\ge -1$

then find

$\mathop{\lim }\limits_{x\to -1} f(x)$

Solution 2. We can take the right hand limit of f(x) as x approaches negative one and that value is equal to negative one because of continuity from the right. We can’t take the left hand limit however because the function is not even defined for x < -1. Since the left and right hand limits don’t match we say that.

$\mathop{\lim }\limits_{x\to -1^{-} } f(x)=DNE \qquad \mathop{\lim }\limits_{x \to -1^{+} } f(x)=-1$