Question

A real number $x$ satisfying $1 - \frac{1}{n}<x \leq 3 + \frac{1}{n}$, for every positive integer $n$, is best described by:

• A. $1<x<4$
• B. $1<x \leq 3$
• C. $0<x \leq 4$
• D. $1 \leq x \leq 3$

Hint:

When solving inequalities involving limits as $n$ approaches infinity, carefully evaluate the boundary conditions to determine the valid range.

Answer 1

The Correct Option is A

We are given that $1 - \frac{1}{n}<x \leq 3 + \frac{1}{n}$ for every positive integer $n$. As $n \to \infty$, we get the limiting values: \[ 1 \leq x \leq 3 \] Thus, the value of $x$ lies between 1 and 4, but never actually reaching 1 or 4. Therefore, the Correct Answer is $1<x<4$.