Question

Find \(\begin{array}{ll}\text { (a) } \lim _{x \rightarrow 0} \frac{1}{x^{3}} & \text { (b) } \lim _{x \rightarrow 0} \frac{1}{x^{6}} \\ \text { (c) } \lim _{x \rightarrow x_{0}} \frac{1}{\left(x-x_{0}\right)^{2 k}} & \text { (d) } \lim _{x \rightarrow x_{0}} \frac{1}{\left(x-x_{0}\right)^{2 k+1}}\end{array}\) \((k=\) positive integer)

Short answer

If so, what is the limit? Answer: The limit does not exist. 2. What is the limit of function (b) \(\frac{1}{x^6}\) as x approaches 0? Answer: The limit is infinity. 3. What is the limit of function (c) \(\frac{1}{(x-x_0)^{2k}}\) as x approaches \(x_0\)? Answer: The limit is infinity. 4. Does the limit of function (d) \(\frac{1}{(x-x_0)^{2k+1}}\) as x approaches \(x_0\) exist? If so, what is the limit? Answer: The limit does not exist.

Step-by-step solution

(a) Limit as x approaches 0 for \(\frac{1}{x^3}\)

In order to find the limit, let's consider both left-hand and right-hand limits of the function. Left-hand limit: \(\lim\limits_{x \rightarrow 0^-} \frac{1}{x^3}\) As x approaches 0 from the left side (negative side), \(x^3\) will have a negative value, and since the power 3 is odd, the denominator will also become negative. Therefore, the overall value of the function will tend to negative infinity. Right-hand limit: \(\lim\limits_{x \rightarrow 0^+} \frac{1}{x^3}\) As x approaches 0 from the right side (positive side), \(x^3\) will have a positive value, and therefore, the denominator will also become positive. So, the value of the function will tend to positive infinity. Since the left-hand limit and right-hand limit are not equal, the limit does not exist for this function.

(b) Limit as x approaches 0 for \(\frac{1}{x^6}\)

Similar to part (a), let's consider the left-hand and right-hand limits of this function. Left-hand limit: \(\lim\limits_{x \rightarrow 0^-} \frac{1}{x^6}\) When x approaches 0 from the left side, \(x^6\) will always be positive, and the function's value will tend to infinity. Right-hand limit: \(\lim\limits_{x \rightarrow 0^+} \frac{1}{x^6}\) As x approaches 0 from the right side, \(x^6\) will also always be positive, meaning the function value will tend to infinity. Since both left-hand and right-hand limits equal to infinity, the limit of the function exists and is equal to infinity.

(c) Limit as x approaches \(x_0\) for \(\frac{1}{(x-x_0)^{2k}}\)

In this function, the denominator is raised to the power of \(2k\), where k is a positive integer. This means that the function will always be nonnegative since any value raised to an even power is nonnegative. As x approaches \(x_0\), the value of the denominator \((x - x_0)^{2k}\) tends to 0, and as a result, the value of the function tends to infinity. Therefore, the limit of the function as x approaches \(x_0\) is infinity.

(d) Limit as x approaches \(x_0\) for \(\frac{1}{(x-x_0)^{2k+1}}\)

In this function, the denominator is raised to the power of \((2k+1)\), where k is a positive integer. This means that, just like in part (a), the denominator will change sign depending on whether x is approaching \(x_0\) from the left or right. Left-hand limit: \(\lim\limits_{x \rightarrow x_0^-} \frac{1}{(x-x_0)^{2k+1}}\) As x approaches \(x_0\) from the left side, the value of \((x - x_0)^{2k+1}\) will be negative, and as a result, the value of the function tends to negative infinity. Right-hand limit: \(\lim\limits_{x \rightarrow x_0^+} \frac{1}{(x-x_0)^{2k+1}}\) When x approaches \(x_0\) from the right side, the value of \((x - x_0)^{2k+1}\) will be positive, and as a result, the value of the function tends to positive infinity. Since the left-hand limit and right-hand limit are not equal, the limit does not exist for this function.

Key concepts

One-Sided Limits

Understanding one-sided limits is crucial when analyzing the behavior of functions as they approach a specific point from different directions. These limits are the values that a function approaches as the variable gets arbitrarily close to a particular point either from the left-side, denoted as \( x \rightarrow c^- \), or from the right-side, denoted as \( x \rightarrow c^+ \).

As seen in the exercise, calculating one-sided limits is essential when the two sides of the limit yield different results. For instance, the one-sided limits for \( \frac{1}{x^3} \) as \( x \) approaches 0 are distinct depending on the direction of approach. From the left (\( x \rightarrow 0^- \) ), the function tends to negative infinity, while from the right (\( x \rightarrow 0^+ \) ), it tends to positive infinity. Since the one-sided limits do not match, the overall limit at that point does not exist.

To improve the student’s comprehension, emphasize the visual interpretation of one-sided limits on a graph and reinforce that discontinuities often require separate left and right-side analysis.

Infinite Limits

An infinite limit describes a situation where a function grows without bound as the variable approaches a certain value. These limits represent scenarios where the function does not settle on a finite number but instead increases (or decreases) towards infinity or negative infinity.

In the provided exercise, parts (a) and (d) showcase infinite limits. For example, the limit \( \frac{1}{x^6} \) as \( x \rightarrow 0 \) is an infinite limit, since whatever the direction we approach 0, the result tends to positive infinity. Infinite limits are indicative of vertical asymptotes in the graph of a function. To enhance understanding, students should be encouraged to graph the functions and observe the vertical asymptotes where the limits are infinite. Highlighting the relationship between the degree of the denominator and the infinity's sign is also very helpful. When the function contains even-powered denominators, as in part (b), the limits on both sides will be the same, either both positive or negative infinite.

Limits at Infinity

Limits at infinity refer to the behavior of functions as the independent variable grows indefinitely in the positive or negative direction. It looks at the long-term behavior of the function as it goes toward positive or negative infinity. These kinds of limits are central to understanding the end behavior of a function and are tied to the concept of horizontal asymptotes.

Considering a function such as \( \frac{1}{(x-x_0)^{2k}} \) as \( x \rightarrow x_0 \), can be seen as an infinite limit. However, if we were to evaluate the limit as \( x \rightarrow \infty \) or \( x \rightarrow -\infty \) for a similar function, we would be examining the limit at infinity. Teaching students to distinguish between infinite limits and limits at infinity is vital, with the former dealing with a function’s behavior near a specific point and the latter assessing the behavior as \( x \) extends beyond all bounds. In-depth examples illustrating how various functions behave as \( x \) becomes very large or very negative could help students grasp the concept of limits at infinity better.