Q. 2 - Calculus

Question

Fill in the blanks to complete each of the following theorem statements: 

$\underset{x\to c}{\lim }f\left(x\right)=L$if and only if $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underline{ }$and$\underset{x\to c^{+}}{\lim }f\left(x\right)=\underline{ }.$

Step-by-Step Solution

Answer

$\underset{x\to c}{\lim }f\left(x\right)=L$ if and only if $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underline{L}$and$\underset{x\to c^{+}}{\lim }f\left(x\right)=\underline{L}.$

Step 1. Given information.

The given incomplete statement is the following. 

$\underset{x\to c}{\lim }f\left(x\right)=L$if and only if $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underline{ }$and$\underset{x\to c^{-}}{\lim }f\left(x\right)=\underline{ } .$

Step 2. Explanation.

If a limit $\underset{x\to c}{\lim }f\left(x\right)$Exists, then the Left and Right Limits Must Exist and Be Equal.

So $\underset{x\to c}{\lim }f\left(x\right)=L$if and only if $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underline{L}$and$\underset{x\to c^{-}}{\lim }f\left(x\right)=\underline{L}.$