Question

Evaluating limits Evaluate the following limits, where \(c\) and \(k\) are constants. $$\lim _{x \rightarrow 2}(5 x-6)^{3 / 2}$$

Short answer

Question: Evaluate the limit of the given function as x approaches 2: \(\lim _{x \rightarrow 2}(5 x-6)^{3 / 2}\) Answer: The limit of the function \((5 x-6)^{3 / 2}\) as x approaches 2 is 8.

Step-by-step solution

Identify the limit notation

The given limit is denoted as \(\lim _{x \rightarrow 2}(5 x-6)^{3 / 2}\). This means we will find the value of the function \((5 x-6)^{3 / 2}\) as \(x\) approaches 2.

Substitute the value of x

Plug in the value of \(x=2\) into the given function: \((5(2)-6)^{3/2}\)

Simplify the expression inside the parentheses

Now, let's simplify the expression inside the parentheses: \((10-6)^{3/2}\)

Continue simplifying the expression

Next, calculate the value inside the parentheses: \((4)^{3/2}\)

Simplify the exponent

Finally, calculate the value by raising 4 to the power of 3/2: \(4^{3/2}=(\sqrt{4})^3=2^3=8\)

Write the final answer

The limit of the function \((5 x-6)^{3 / 2}\) as \(x\) approaches 2 is 8: $$\lim _{x \rightarrow 2}(5 x-6)^{3 / 2}=8$$