Q. 13 - Calculus

Question

The Squeeze Theorem for Limits: If $I\left(x\right)\le f\left(x\right)\le u\left(x\right)$ for all $x$sufficiently close to ?, but not necessarily at ?, and if $\underset{x\to c}{\lim }l\left(x\right)\text{ and }\underset{x\to c}{\lim }u\left(x\right)$are both equal to L, then ?. 

Step-by-Step Solution

Answer

If $I\left(x\right)\le f\left(x\right)\le u\left(x\right)$ for all $x$ sufficiently close to $c$ but not necessarily at $x=c,$ and if $\underset{x\to c}{\lim }I\left(x\right)\text{ and }\underset{x\to c}{\lim }u\left(x\right)$ are both equal to L, then, $\underset{x\to c}{\lim }f\left(x\right)=L.$

Step 1. Given Information.

The given statement is,

If $I\left(x\right)\le f\left(x\right)\le u\left(x\right)$for all $x$ sufficiently close to ?, but not necessarily at ?, and if $\underset{x\to c}{\lim }I\left(x\right)\text{ and }\underset{x\to c}{\lim }u\left(x\right)$ are both equal to $L,$ then $\underset{x\to c}{\lim }f\left(x\right)=L$

Step 2. Fill the blanks.

According to the Squeeze Theorem for Limits,

If $\left(x\right)\le f\left(x\right)\le u\left(x\right)$ for all $x$ sufficiently close to $c,$but not necessarily at $x=c,$and if $\underset{x\to c}{\lim }I\left(x\right)\text{ and }\underset{x\to c}{\lim }u\left(x\right)$are both equal to $L,$ then $\underset{x\to c}{\lim }f\left(x\right)=L$.