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Magazine Chapter 2: Problem 26
Decide which of the given one-sided or two-sided limits exist as numbers, which as \(\infty\), which as \(-\infty\), and which do not exist. Where the limit is a number, evaluate it. $$ \lim _{x \rightarrow 0} \frac{x}{|x|} $$
Short Answer
Expert verified
The two-sided limit does not exist, as the right and left limits are different.
Step by step solution
01
Understand the Function
Examine the given function \( \frac{x}{|x|} \). This function depends on the sign of \( x \) because it involves the absolute value of \( x \). Check how the function behaves for \( x > 0 \), \( x < 0 \), and \( x = 0 \).
02
Define the Function for Different x
For \( x > 0 \), \( |x| = x \) so \( \frac{x}{|x|} = \frac{x}{x} = 1 \). For \( x < 0 \), \( |x| = -x \) so \( \frac{x}{|x|} = \frac{x}{-x} = -1 \). The function is undefined at \( x = 0 \) since \( \frac{0}{|0|} \) is undefined.
03
Calculate Right-Hand Limit
Compute the limit as \( x \) approaches 0 from the right: \( \ lim_{x \rightarrow 0^+} \frac{x}{|x|} = \lim_{x \rightarrow 0^+} 1 = 1 \). This is because for positive \( x \), \( \frac{x}{|x|} = 1 \).
04
Calculate Left-Hand Limit
Compute the limit as \( x \) approaches 0 from the left: \( \ lim_{x \rightarrow 0^-} \frac{x}{|x|} = \lim_{x \rightarrow 0^-} -1 = -1 \). This is because for negative \( x \), \( \frac{x}{|x|} = -1 \).
05
Determine Overall Limit
Since the right-hand limit (1) does not equal the left-hand limit (-1), the two-sided limit \( \lim_{x \rightarrow 0} \frac{x}{|x|} \) does not exist.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
One-sided Limits
When we talk about one-sided limits, we're looking at how a function behaves as the input approaches a specific point from just one side. There are two types of one-sided limits: left-hand and right-hand limits.
To further realize the importance of one-sided limits, consider the function given in the exercise, \( \lim_{x \rightarrow 0} \frac{x}{|x|} \). Calculating the one-sided limits helps us understand the behavior on either side of zero.
To further realize the importance of one-sided limits, consider the function given in the exercise, \( \lim_{x \rightarrow 0} \frac{x}{|x|} \). Calculating the one-sided limits helps us understand the behavior on either side of zero.
- **Right-hand limit**: This is noted by \( x \rightarrow 0^+ \), meaning we're approaching 0 from values greater than 0.
- **Left-hand limit**: Denoted by \( x \rightarrow 0^- \), indicating approach from values less than 0.
Two-sided Limits
Two-sided limits look at how a function behaves as it approaches a specific point from both left and right directions. This implies that both the left-hand and right-hand limits match to have a meaningful two-sided limit.
Concerning our example, the function \( \lim_{x \rightarrow 0} \frac{x}{|x|} \) shows different behaviors as \( x \) approaches 0 from the positive and negative sides.
Concerning our example, the function \( \lim_{x \rightarrow 0} \frac{x}{|x|} \) shows different behaviors as \( x \) approaches 0 from the positive and negative sides.
- Since the right-hand limit is \( 1 \) and the left-hand limit is \( -1 \), they do not match.
- When these one-sided limits are not equal, the two-sided limit is considered to **not exist**.
Absolute Value
Absolute value, denoted by \(|x|\), is a fundamental concept where the value of \(x\) is always considered positive, without regard to its sign. It's like asking for the distance of a number from zero.
In the context of the function \( \frac{x}{|x|} \), the absolute value plays a significant role:
In the context of the function \( \frac{x}{|x|} \), the absolute value plays a significant role:
- For \(x > 0\), \(|x| = x\) meaning the fraction becomes \(1\).
- For \(x < 0\), \(|x| = -x\) turning \( \frac{x}{|x|} \) into \(-1\).
Undefined Function
An undefined function occurs when you attempt a calculation that does not result in a real number or defined output. Most frequently, undefined expressions include division by zero.
In the worked example, at \(x = 0\), the function \( \frac{x}{|x|} \) becomes undefined:
In the worked example, at \(x = 0\), the function \( \frac{x}{|x|} \) becomes undefined:
- The calculation \( \frac{0}{|0|} \) is not possible, as it leads to division by zero.
- Thus, the function doesn't provide a valid output at \(x = 0\), but still holds meaningful behavior towards the sides of zero through one-sided limits.
Left-hand Limit
The left-hand limit examines the behavior of a function as the input approaches a specific number from the left side or lower values. It's represented as \( \lim_{x \rightarrow c^-} f(x) \).
When calculating for the function \( \lim_{x \rightarrow 0^-} \frac{x}{|x|} \):
When calculating for the function \( \lim_{x \rightarrow 0^-} \frac{x}{|x|} \):
- As \(x\) nears 0 from negative values, \(\frac{x}{|x|}\) simplifies to \(-1\).
- This demonstrates that the left-hand limit at zero for this function equals \(-1\).
Right-hand Limit
The right-hand limit looks at how a function behaves as an input approaches a specific value from the right side or higher values. It's denoted as \( \lim_{x \rightarrow c^+} f(x) \).
Regarding the worked example function \( \lim_{x \rightarrow 0^+} \frac{x}{|x|} \):
Regarding the worked example function \( \lim_{x \rightarrow 0^+} \frac{x}{|x|} \):
- With \(x\) getting close to 0 from positive values, \(\frac{x}{|x|}\) simplifies to \(1\).
- This proves that the right-hand limit at zero for this function is \(1\).
