Question

True or False? In Exercises \(115-120\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \lim _{x \rightarrow 0} \frac{|x|}{x}=1 $$

Short answer

The statement is False. The limit does not exist because the left-hand limit and the right-hand limit are not equal.

Step-by-step solution

Understand the function

The function here is \( f(x) = \frac{|x|}{x} \) which means that when x is positive, \( f(x)=1 \) and when x is negative, \( f(x)=-1 \). This is due to the properties of the absolute value function: |x| is x when x is positive or zero and -x when x is negative.

Evaluate the limit

Determining a limit involves evaluating the function at values that approach the number. In this case, x is approaching 0. Hence we need to consider values of x from both the left-hand side (negative values) and the right-hand side (positive values). However, since the values of the function differ for positive and negative x (as explained in step 1), the limit from the positive side is not equal to the limit from the negative side.

Final step

Recall that a limit only exists if the left-hand limit and the right-hand limit are equal. In this case, they are not equal, the limit does not exist. Hence, the statement \( \lim _{x \rightarrow 0} \frac{|x|}{x}=1 \) is False.

Key concepts

Absolute Value Function

The absolute value function, denoted as \(|x|\), is a fundamental concept in mathematics that returns the non-negative value of a given number, irrespective of its sign. In simpler terms, it measures the distance from zero on the number line, making sure the result is always positive.

For any real number \(x\):

• If \(x\) is positive or zero, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\), which is equivalent to taking the opposite of a negative number to make it positive.

In the exercise provided, the function \(f(x) = \frac{|x|}{x}\) demonstrates how the absolute value influences the result based on the sign of \(x\). When \(x > 0\), the absolute value of \(x\) equals \(x\), so \(f(x) = 1\). Conversely, if \(x < 0\), the absolute value converts \(x\) to \(-x\), making \(f(x) = -1\). This behavior is essential for understanding the determination of limits as \(x\) approaches zero from different directions.

One-Sided Limits

One-sided limits are crucial when a function shows different behaviors from the left or right side of a particular point. Instead of considering both sides simultaneously, one-sided limits assess the value the function approaches from one specific direction.

For the function \(f(x) = \frac{|x|}{x}\), as \(x\) nears zero, it's essential to inspect separate scenarios:

• Left-Hand Limit (LHL): As \(x\) approaches 0 from the negative side \((-x)\), \(f(x) = -1\).
• Right-Hand Limit (RHL): As \(x\) approaches 0 from the positive side \((+x)\), \(f(x) = 1\).

Recognizing these distinctions is important when determining the overall limit, as the limit only exists when both the left-hand and right-hand limits agree. For this particular problem, since \(f(x)\) results in two different values as \(x\) approaches zero from each side, the overall limit does not exist.

Limit Does Not Exist

In calculus, determining whether a limit exists at a certain point involves comparing the function's behavior approaching that point from the left and the right. A limit \(\lim_{x \to a} f(x)\) is said to exist if both the left-hand limit \((LHL)\) and the right-hand limit \((RHL)\) are equal at \(x = a\).

For \(f(x) = \frac{|x|}{x}\), as \(x\) nears 0:

• The LHL is \(-1\) as \(x\) approaches from the negative direction.
• The RHL is \(1\) as \(x\) approaches from the positive direction.

Since these values are not the same, the limit \(\lim_{x \to 0} \frac{|x|}{x} = 1\) does not exist. This disproves the statement in the exercise, showing that simply acknowledging the function's distinct behaviors from separate directions is essential in understanding limits.

Continuous vs. Discontinuous Functions

Continuous functions have no breaks, jumps, or holes in their graphs. This means they have the same input-output behavior approaching a particular point from any direction. A function is continuous at a point \(x = a\) if three conditions are satisfied:

• The function \(f(x)\) is defined at \(x = a\).
• The limit \(\lim_{x \to a} f(x)\) exists.
• The value of the function at \(a\) equals the limit of the function as \(x\) approaches \(a\): \(f(a) = \lim_{x \to a} f(x)\).

Discontinuous functions, like \(f(x) = \frac{|x|}{x}\) at \(x = 0\), fail one or more of these conditions. In our problem, the function changes output sharply from \(-1\) to \(1\) as \(x\) moves across zero, indicating a jump discontinuity. The abrupt shift showcases a fundamental aspect of discontinuous functions: they lack the smooth transition seen in continuous ones, which disrupts the existence of a limit at specific points.