Chapter 13.1, Problem 2E

We write $\underset{x\to a}{\lim }f(x)=L$ and say that the ______ of f(x) as x approaches a from the ______ (left/right) is equal to _______. To find the left-hand limit, we try values for x that are _______ (less/greater) than a. A limit exists if and only if both the _______-hand and _______-hand limits exist and are _______.

To determine

To fill: The blanks in the statement “We write $\underset{x\to a^{-}}{\lim }f\left(x\right)=L$ and say that the _____ of $f\left(x\right)$ as x approaches $a$ from the _____ (left/right) is equal to _____. To find the left-hand limit, we try values for $x$ that are _____ (less/greater) than a. A limit exists if and only if both the _____-hand and _____-hand limit exists and are _____.”

Answer to Problem 2E

We write $\underset{x\to a^{-}}{\lim }f\left(x\right)=L$ and say that the left-hand limit of $f\left(x\right)$ as x approaches $a$ from the left is equal to L. To find the left-hand limit, we try values for $x$ that are less than a. A limit exists if and only if both the left-hand and right-hand limit exists and are equal.

Explanation of Solution

Explanation:

The given limit is, $\underset{x\to a^{-}}{\lim }f\left(x\right)=L$.

The symbol $x\to a^{-}$ indicates that, x approaches the values which are to the left of a.

That is, while measured on a number line, x approaches a from the left side.

Therefore, only the values of x that are less than a are considered.

Thus, by the definition of one-sided limit, $\underset{x\to a^{-}}{\lim }f\left(x\right)=L$ is read as the left-hand limit of $f\left(x\right)$ as x approaches a from the left is equal to L.

Similarly, right-hand limit can also be defined.

The right-hand limit of $f\left(x\right)$ as x approaches $a$ from the right is equal to L, is written as $\underset{x\to a^{+}}{\lim }f\left(x\right)=L$.

The limit $\underset{x\to a}{\lim }f\left(x\right)=L$, if and only if the left-hand and right-hand limits exists and are equal.

That is, $\underset{x\to a^{-}}{\lim }f\left(x\right)=L\text{ and}\underset{x\to a^{+}}{\lim }f\left(x\right)=L$.