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Chapter 3: Problem 23

Find \(\lim _{x \rightarrow a^{-}} f(x), \lim _{x \rightarrow a^{+}} f(x),\) and \(\lim _{x \rightarrow a} f(x)\) at the indicated value for the indicated function. Do not use a computer or graphing calculator. $$ a=1, f(x)=\frac{1}{x-1} $$

Short Answer

Expert verified
Left-hand limit: \(-\infty\), right-hand limit: \(+\infty\), limit does not exist.

Step by step solution

01

Evaluate the Left-Hand Limit

To find the left-hand limit \( \lim _{x \rightarrow 1^{-}} \frac{1}{x-1} \), we consider values of \( x \) that approach 1 from the left. These values are slightly less than 1 (i.e., \( x < 1 \)). As \( x \) gets closer to 1 from the left, \( x-1 \) becomes a small negative number. Dividing 1 by a small negative number results in a large negative value. Therefore, \( \lim _{x \rightarrow 1^{-}} \frac{1}{x-1} = -\infty \).
02

Evaluate the Right-Hand Limit

To find the right-hand limit \( \lim _{x \rightarrow 1^{+}} \frac{1}{x-1} \), we consider values of \( x \) that approach 1 from the right. These values are slightly greater than 1 (i.e., \( x > 1 \)). As \( x \) gets closer to 1 from the right, \( x-1 \) becomes a small positive number. Dividing 1 by a small positive number results in a large positive value. Therefore, \( \lim _{x \rightarrow 1^{+}} \frac{1}{x-1} = +\infty \).
03

Compare Left and Right Limits

To find \( \lim _{x \rightarrow 1} \frac{1}{x-1} \), we compare the left-hand and right-hand limits. We have \( \lim _{x \rightarrow 1^{-}} \frac{1}{x-1} = -\infty \) and \( \lim _{x \rightarrow 1^{+}} \frac{1}{x-1} = +\infty \). Since these two limits are not equal, \( \lim _{x \rightarrow 1} \frac{1}{x-1} \) does not exist.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Left-Hand Limit
The concept of left-hand limits is essential when exploring the behavior of a function as it approaches a specific point from the left side along the x-axis. For example, consider the left-hand limit of the function \(f(x) = \frac{1}{x-1}\)\ as \(x\) approaches 1, symbolically written as \(\lim_{x \rightarrow 1^{-}} \frac{1}{x-1}\).

Here鈥檚 a simple way to think about it:
  • As \( x \) gets closer to 1 from values less than 1 (e.g., 0.9, 0.99), \(x-1\) turns into a small negative number.
  • Dividing 1 by a tiny negative number yields a very large negative value.
Thus, when approaching 1 from the left, the function \(\frac{1}{x-1}\) diminishes towards \(-\infty\).

The left-hand limit offers a glimpse of the function's behavior just before it reaches the critical point of interest, often revealing discontinuities or vertical asymptotes that a simple evaluation might miss.
Right-Hand Limit
Understanding the right-hand limit involves looking at how a function behaves as it approaches a certain point from the right side along the x-axis. To illustrate, consider the function \(f(x) = \frac{1}{x-1}\)\ as \(x\) approaches 1, written as \(\lim_{x \rightarrow 1^{+}} \frac{1}{x-1}\).

This concept is easier to grasp with the following steps:
  • When \(x\) approaches 1 from values greater than 1 (like 1.1, 1.01), \(x-1\) is transformed into a small positive number.
  • Dividing 1 by a minuscule positive number results in a massively large positive number.
Therefore, as \(x\) nears 1 from the right, the expression \(\frac{1}{x-1}\) reaches \(+\infty\).

The right-hand limit is vital for understanding how functions behave as they approach key points, frequently revealing rapid growth or infinite tendencies that might not be obvious at first glance.
Undefined Limit
An undefined limit occurs when the left-hand limit and the right-hand limit at a given point do not match. Let's use the function \(f(x) = \frac{1}{x-1}\)\ as it approaches 1, noted as \(\lim_{x \rightarrow 1} \frac{1}{x-1}\).

To break it down:
  • The left-hand limit as \(x\) approaches 1 is \(-\infty\), as discussed before.
  • The right-hand limit as \(x\) nears 1 is \(+\infty\).
  • Since \(-\infty\) and \(+\infty\) are not equal, the overall limit is undefined.
In calculus, when these two one-sided limits diverge, the conclusion is that the function does not settle to a single value or behavior at that point.

An undefined limit signifies a profound discontinuity at the point of interest, emphasizing the need to carefully assess such points when analyzing functions, as they indicate behaviors like jumps, rapid changes, or breaks in the graph of a function.

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Most popular questions from this chapter

Environmental Entomology Allen and coworkers \(^{40}\) found the relationship between the temperature and the number of eggs laid by the female citrus rust mite. They found that if \(x\) is the temperature in degrees Celsius and \(y\) is the total eggs per female, \(y\) was given approximately by the equation \(-0.00574 x^{3}+0.292 x^{2}-3.632 x+11.661\) Graph using a window with dimensions [10,38.2] by \([0,20] .\) Have your computer or graphing calculator draw tangent lines to the curve when \(x\) is \(19,23.5,25,28,\) and 31\. Note the slope and relate this to the rate of change. Interpret what each of these numbers means. On the basis of this model, describe what happens to the rate of change of number of eggs as temperature increases.

Determine graphically and numerically whether or not any of the following limits exist. a. \(\lim _{r \rightarrow 0}\left(2^{-|x|}\right)\) b. \(\lim _{x \rightarrow 0^{+}}\left(x^{2 x}\right)\) c. \(\lim _{x \rightarrow 0^{+}}(x \ln x)\) d. \(\lim _{x \rightarrow 0^{+}}\left(\ln \frac{1}{x}\right)^{x}\) e. \(\lim _{x \rightarrow 1}\left(\frac{x-1}{\ln x}\right)\)

Find, without graphing, where each of the given functions is continuous. $$ \frac{x-1}{x^{2}+1} $$

Finance A study of Dutch manufacturers \(^{38}\) found that the total cost \(C\) in thousands of guilders incurred by a company for hiring (or firing) \(x\) workers was approximated by \(C=0.0071 x^{2} .\) Find the rate of change of costs with respect to workers hired when 100 workers are hired. Give units and interpret your answer.

Population Ecology Lactin and colleagues \(^{42}\) collected data relating the feeding rate in \(y\) units of the second-instar Colorado potato beetle and the temperature \(T\) in degree Celsius. They found that the equation \(y=-0.0239 T^{2}+\) \(1.3582 T-14.12\) was approximately true. Graph using a window with dimensions [10,57] by \([0,7] .\) Have your computer or graphing calculator draw tangent lines to the curve when \(x\) is \(20,25,29,32,\) and \(38 .\) Note the slope and relate this to the rate of change. Interpret what each of these numbers means. On the basis of this model, describe what happens to the rate of change of feeding rate as temperature increases.
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