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Chapter 2: Problem 61

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

Short Answer

Expert verified
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

01

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.
02

Substitute the value of x

Plug in the value of \(x=2\) into the given function: \((5(2)-6)^{3/2}\)
03

Simplify the expression inside the parentheses

Now, let's simplify the expression inside the parentheses: \((10-6)^{3/2}\)
04

Continue simplifying the expression

Next, calculate the value inside the parentheses: \((4)^{3/2}\)
05

Simplify the exponent

Finally, calculate the value by raising 4 to the power of 3/2: \(4^{3/2}=(\sqrt{4})^3=2^3=8\)
06

Write the final answer

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

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Most popular questions from this chapter

Let $$g(x)=\left\\{\begin{array}{ll}x^{2}+x & \text { if } x<1 \\\a & \text { if } x=1 \\\3 x+5 & \text { if } x>1.\end{array}\right.$$ a. Determine the value of \(a\) for which \(g\) is continuous from the left at 1. b. Determine the value of \(a\) for which \(g\) is continuous from the right at 1. c. Is there a value of \(a\) for which \(g\) is continuous at \(1 ?\) Explain.

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