Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \sqrt[3]{x}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Cube Root Function

The function given is the cube root function, expressed as \( f(x) = \sqrt[3]{x} \). This function is defined for all real numbers, which means it can take any real number input, including fractions and negatives.

Step 2: Evaluate the Left-Hand Limit

Consider the behavior of \( \sqrt[3]{x} \) as \( x \) approaches 0 from the left (values of \( x \) that are negative). For negative values close to 0, the cube root of \( x \) will also be close to 0, approaching 0 as \( x \) approaches 0.

Step 3: Evaluate the Right-Hand Limit

Now consider the behavior of \( \sqrt[3]{x} \) as \( x \) approaches 0 from the right (positive values of \( x \)). For positive values of \( x \) close to 0, the cube root will again be close to 0. It approaches 0 as \( x \) approaches 0 from the right.

Step 4: Combine the Left and Right-Hand Limits

Since both the left-hand limit and the right-hand limit of \( f(x) = \sqrt[3]{x} \) as \( x \) approaches 0 have the same result, we conclude that the overall limit exists. Both limits approach 0, thus:

Step 5: Conclude the Limit

Based on the analysis of the left and right-hand limits, the limit exists and is equal to 0. Therefore: \[ \lim_{x \rightarrow 0} \sqrt[3]{x} = 0 \]

Key Concepts

Cube Root Function

The cube root function is an important concept in calculus. It's represented as \( f(x) = \sqrt[3]{x} \). Unlike the square root, the cube root can handle all real numbers, including negative ones. This means it's defined everywhere on the real number line.

• The domain includes negative, positive, and zero values.
• For positive numbers, the cube root is positive.
• For negative numbers, the cube root remains negative.

So, when you're working with a cube root, you're in a very flexible mathematical situation where you don't have to worry about undefined values like you do with even roots.

Left-Hand Limit

The concept of a left-hand limit is about observing how a function behaves as you approach a specific point from the left.

• For the cube root function \( \sqrt[3]{x} \), the left-hand limit as \( x \) approaches 0 means we're looking at values just smaller than 0.
• Since we approach from negative values, the outcomes also stay negative but tend towards 0.
• This means \( \lim_{x \to 0^-} \sqrt[3]{x} = 0 \).

Understanding the left-hand limit helps build the complete picture of the function's behavior around the point of interest, in this case, 0.

Right-Hand Limit

A right-hand limit observes how a function behaves when approaching a point from the right.

• In the case of \( \sqrt[3]{x} \), we're examining values slightly greater than 0.
• These values are positive but very small and the cube root also approaches 0.
• Thus, as \( x \to 0^+ \), we find that \( \lim_{x \to 0^+} \sqrt[3]{x} = 0 \).

Evaluating right-hand limits allows us to see how the function trends as it nears exact points from one direction. Together with the left-hand limit, these observations are key to fully understanding limits.

Existence of Limits

The existence of limits is a cornerstone in calculus, determining whether a function approaches a particular value at a certain point.

• Both the left-hand and right-hand limits must reach the same value for a limit to truly exist at a point.
• In our exercise, since both halves approached 0, the overall limit exists.
• This means \( \lim_{x \to 0} \sqrt[3]{x} = 0 \), confirming the cube root function behaves predictably as \( x \) nears 0 from both sides.

Hence, checking both directions is crucial. If they differ, the limit does not exist. This analysis forms the foundation for more complex calculus studies by assuring what's happening in a close neighborhood around a point.