Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 3^{+}} \frac{|x-3|}{x-3} $$

Short answer

The limit of the given function as \(x\) approaches 3 from the right side is 1.

Step-by-step solution

Understand the absolute value function

The absolute value of \(x-3\) is equal to \(x-3\) itself when \(x\) is greater than 3, and it is equal to \(-(x-3)\) when \(x\) is less than 3. However, since we are finding the limit as \(x\) approaches 3 from the right side, we consider \(x\) to be greater than 3. Therefore, for \(x > 3\), the absolute value function \(|x-3|\) can be expressed as \(x-3\).

Substitute the absolute value function

Substitute the function \(|x-3|\) with \(x-3\) in the expression, and we have: \(\lim _{x \rightarrow 3^{+}} \frac{x-3}{x-3} \).

Simplify the expression

Simplify the expression \(\frac{x-3}{x-3}\) to 1, the limit as \(x\) approaches 3 from the right side becomes: \(\lim _{x \rightarrow 3^{+}} 1\).

Compute the limit

The limit of a constant function is equal to the constant itself, regardless of the value that \(x\) approaches. So, \(\lim _{x \rightarrow 3^{+}} 1 = 1\).