Question

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow 2 } \sqrt { x + 3 }$$

Short answer

The limit of the function \(f(x) = \sqrt{x+3}\) as \(x\) approaches 2 is \(\sqrt{5}\).

Step-by-step solution

Determine the function

Identify the function that we are taking the limit of, which is \(f(x) = \sqrt{x+3}\). The task is to determine what \(f(x)\) approaches as \(x\) approaches 2. That is, we are asked to find \(\lim _ { x \rightarrow 2 } f(x)\).

Substitute the limit value

As the limit value is \(x=2\), we substitute this into the function to get \(f(2)\). As such, \(f(2) = \sqrt{2+3}\).

Simplify the equation

Simplify \(f(2)\) to obtain a numerical value. \(f(2) = \sqrt{5}\).

Graphical representation

While not explicitly presented here, the graphing of the function \(f(x) = \sqrt{x+3}\) should reveal that as \(x\) approaches 2, the value of \(f(x)\) approaches \(\sqrt{5}\), as found in step 3.