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Chapter 9: Problem 20

Find the limit. \(\lim _{x \rightarrow 0^{-}}\left(x^{2}-\frac{1}{x}\right)\)

Short Answer

Expert verified
The limit of the given function as \(x\) approaches 0 from the left is infinity.

Step by step solution

01

Break the function into two limits

We will start by separating the limit into two parts using the property that the limit of a sum or difference is the sum or difference of the limits: \(\lim _{x \rightarrow 0^{-}}\left(x^{2}-\frac{1}{x}\right) = \lim _{x \rightarrow 0^{-}}x^{2} - \lim _{x \rightarrow 0^{-}}\frac{1}{x}\)
02

Evaluate the limit of each term

Now we will find the limit of each term separately. For the first term, \(x^{2}\), as \(x\) goes towards 0, \(x^{2}\) will also go towards zero, so \(\lim _{x \rightarrow 0^{-}}x^{2} = 0\). However, the second term, \(\frac{1}{x}\), is undefined as \(x\) approaches 0. For \(x \rightarrow 0^-\), the value of \(\frac{1}{x}\) goes to \(-\infty\). Therefore, \(\lim _{x \rightarrow 0^{-}}\frac{1}{x} = -\infty\)
03

Combine the two solved limits

Finally, we combine the two solved limits to find our solution: \(0 - (-\infty) = \infty\)

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Most popular questions from this chapter

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=\left\\{\begin{array}{l}x^{2}+1, x \leq 0 \\ 1-2 x, x>0\end{array}\right.\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-x^{2}-2 x+3\)

The variable cost for the production of a calculator is \(\$ 14.25\) and the initial investment is \(\$ 110,000\). Find the total cost \(C\) as a function of \(x\), the number of units produced. Then use differentials to approximate the change in the cost for a one-unit increase in production when \(x=50,000\). Make a sketch showing \(d C\) and \(\Delta C\). Explain why \(d C=\Delta C\) in this problem.

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