Chapter 13, Problem 1RCC

1. (a) Explain what is meant by lim_x→a f(x) = L.
2. (b) If lim_x→2 f(x) = 5, is it possible that f(2) = 3?
3. (c) Find lim_x→2 x².

To determine

To Explain: The expression ${\lim }_{x\to a}f\left(x\right)=L$.

Explanation of Solution

The expression ${\lim }_{x\to a}f\left(x\right)=L$ is read as the limit of the function approaches to the number $L$ when x is very close to a but not equal to a.

The above expression define as the limit of the function is very close to the number $L$ but not exactly equal to $L$.

To determine

To verify: Whether the expression $f\left(2\right)=3$ is possible.

Explanation of Solution

The given limit of the function at the number $x=2$ is ${\lim }_{x\to 2}f\left(x\right)=5$.

The limit of the function states that the value of $f\left(x\right)$ is very close to the number $L$ but not exactly equal to $L$, when x is very close to a but not equal to a.

Here, the limit of the function at $x=2$ is 5.

The given expression is,

${\lim }_{x\to 2}f\left(x\right)=5$

Substitute 2 for x in the above equation,

$f\left(2\right)=5$

So, the limit of the function at $x=2$ is 5.

Thus, $f\left(2\right)=3$ is not possible for the given limit of the function.

To determine

To find: The value of the expression ${\lim }_{x\to 2}{x}^2$.

Answer to Problem 1RCC

The value of the given expression ${\lim }_{x\to 2}{x}^2$ is 4.

Explanation of Solution

Given:

The limit of the function is ${\lim }_{x\to 2}{x}^2$.

Calculation:

The expression is,

${\lim }_{x\to 2}{x}^2$

Substitute 2 for x in the above expression,

$\begin{array}{c}{\lim }_{x\to 2}{x}^2={\left(2\right)}^2\\ =4\end{array}$

Thus, the value of the given expression ${\lim }_{x\to 2}{x}^2$ is 4.