Question

In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow 0^{-}}\left(x^{2}-\frac{2}{x}\right) $$

Short answer

The limit of the function \(x^{2}-\frac{2}{x}\) as \(x\) approaches \(0\) from the negative side is \(-\infty\).

Step-by-step solution

Approach \(x^{2}\) and \(-\frac{2}{x}\) individually

Decompose the given function into two separate functions: \(x^{2}\) and \(-\frac{2}{x}\). We will look at these functions and their behavior as \(x\) approaches \(0\) from the negative side separately.

Apply the limit to the first term

To find \(\lim _{x \rightarrow 0^{-}} x^{2}\) for the first term, we use the property that the limit is equal to the function's value at the given point if that function is continuous at that point. We note that a polynomial function, such as \(x^{2}\), is continuous everywhere, therefore the limit at \(0\) is just \(0^{2} = 0\). So, \(\lim _{x \rightarrow 0^{-}} x^{2} = 0\).

Apply the limit to the second term

Regarding the second term, \(\lim _{x \rightarrow 0^{-}} \left(-\frac{2}{x}\right)\), we recognize that this term is not defined when \(x=0\). But because we're approaching from negative values, this means the term will tend to \(-\infty\).

Combine results

The given function is the sum of the two terms we had analyzed. Combining the results, we find our final answer: \(\lim _{x \rightarrow 0^{-}}\left(x^{2}-\frac{2}{x}\right) = 0 - \infty = -\infty\).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is divided into two main parts: differential calculus, which concerns itself with the concept of a derivative, and integral calculus, which focuses on the concept of an integral. The limit is a fundamental concept in calculus, used to describe the behavior of functions as they approach a specific point or infinity.

For example, in the exercise \( \lim _{x \rightarrow 0^{-}}\left(x^{2}-\frac{2}{x}\right) \), we are asked to find the limit of a function as \( x \) approaches 0 from the left side. The solution requires us to understand how each component of the function behaves individually as \( x \) approaches 0, and then combine those behaviors to find the overall limit of the function.

Continuity

Continuity is a property of a function that describes whether the function is free of interruptions or jumps. A function is said to be continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point. This is known as the limit being \(\text{finite}\) and \(\text{equal}\) to the function value.

In our exercise, \( x^{2} \) is a polynomial function, which is known to be continuous at every point on the real number line. Hence, as \( x \) approaches 0 from the left \( \lim _{x \rightarrow 0^{-}} x^{2} = 0^{2} = 0 \), complying with the continuity of polynomial functions. By understanding continuity, we can quickly conclude this part of the limit problem without further calculation.

Approaching Infinity

The concept of approaching infinity \( (\infty) \) is used when the values of a function increase or decrease without bound as the variable approaches a certain value or infinity itself. In the provided exercise, when evaluating \( \lim _{x \rightarrow 0^{-}} \left(-\frac{2}{x}\right) \) we see that as \( x \) gets closer to zero from the left \( (x < 0) \) the value of \( -\frac{2}{x} \) gets more and more negative without any limit. This behavior is indicative of the term tending towards negative infinity \( (-\infty) \).

This concept is essential when dealing with functions that are not continuous at a certain point, which is a stark contrast to the continuity explained in the previous section. Calculus provides the tools to handle such cases where simple substitution is not enough to find a limit.

Polynomial Functions

Polynomial functions are expressions that involve coefficients and variables raised to non-negative integer powers. A key characteristic of polynomial functions is that they are continuous and smooth everywhere, which makes them relatively straightforward to work with in terms of limits and calculus in general.

In our step by step solution, \( x^{2} \) is a polynomial and its behavior as \( x \) approaches any real number is predictable - it will simply be the value of \( x \) squared. This predictability comes from the non-negative integer powers in polynomial functions, which ensure that there are no points of discontinuity. Understanding the nature of polynomial functions often simplifies the process of finding limits, as we saw in the exercise where the limit of \( x^{2} \) as \( x \) approaches 0 can be quickly determined.