Question

Determining a Limit In Exercises 71 and \(72,\) consider the function \(f(x)=\sqrt{x} .\) Is $$\lim _{x \rightarrow 0} \sqrt{x}=0$$ a true statement? Explain.

Short answer

Yes, \(\lim _{x \rightarrow 0^{+}} \sqrt{x} = 0\) is a true statement. It's important to note, however, this is a one-sided limit because the square root function is only defined for non-negative values of \(x\).

Step-by-step solution

Understanding the Limit Notation

The given expression \(\lim _{x \rightarrow 0} \sqrt{x}\) is asking what the function \(f(x) = \sqrt{x}\) approaches as \(x\) gets closer and closer to 0. In other words, it's interested in the behaviour of the function near 0, not at the exact point where x is equal to 0.

Calculating the Limit

For a positive x getting closer to 0, the function \(f(x) = \sqrt{x}\) also becomes smaller and smaller, approaching 0. So, the limit of this function as\(x\) approaches 0 from the right is indeed 0.

Evaluating the Limit

However, the square root function is undefined for negative values of \(x\). Hence, we can't approach 0 from the left. We don't have to worry about the functions behaviour from the left because it doesn't exist there. As \(x\) approaches 0 from the right (positive side), the limit is 0.

Key concepts

Square Root Function

The square root function, represented by \(f(x) = \sqrt{x}\), is a fundamental concept in mathematics. It is defined only for non-negative numbers, meaning \(x\) has to be 0 or greater. The square root of a positive number \(x\) is a number that, when multiplied by itself, gives \(x\) as the result.
For example:

• The square root of 4 is 2, since \(2 \times 2 = 4\).
• Similarly, the square root of 9 is 3, because \(3 \times 3 = 9\).

What's interesting about the square root function is its graphical representation. When plotted on a graph, it forms a distinctive curve known as a parabola that opens towards the right. It starts at the origin (0,0) and curves gently upwards to the right. The growth of this function becomes slower as \(x\) grows larger, but it continues indefinitely without ever leveling off. This makes the square root function increasing and non-linear, which is crucial for understanding its behavior in limit scenarios.

Limit Notation

Limit notation is a crucial part of calculus, used to understand the behavior of functions as they approach a specific point. In the expression \(\lim_{x \to 0} \sqrt{x}\), we use limit notation to ask, "What does \(\sqrt{x}\) approach as \(x\) gets infinitely close to 0?"
This notation involves a few parts:

• "\(\lim\)" denotes the limit operation.
• "\(x \to 0\)" indicates that \(x\) is approaching the value 0.
• The expression \(\sqrt{x}\) signifies the function under consideration.

The critical idea is that the limit helps us understand tendencies rather than exact values at a point. Even if \(x = 0\) might be undefined in some cases (like square root of negative numbers), limit notation helps us explore what's happening as \(x\) gets near that value without actually reaching it. This is particularly useful in cases where direct evaluation is challenging or impossible.

Behavior of Functions Near a Point

Understanding the behavior of functions near a point is essential in learning about limits. For the square root function \(f(x) = \sqrt{x}\), we are interested in how the function behaves as \(x\) approaches a specific value, in this case, 0.
The behavior near a point involves examining what occurs as \(x\) approaches from one side or both sides:

• Approaching from the right (positive side): For \(x > 0\), as \(x\) becomes smaller, \(\sqrt{x}\) also becomes smaller. As \(x\) nears 0, the value of \(\sqrt{x}\) approaches 0.
• Approaching from the left (negative side): The function \(\sqrt{x}\) is undefined for negative \(x\), so we do not consider movement from this direction.

By focusing only on the right side, the behavior of \(\sqrt{x}\) as \(x\) approaches 0 once more confirms that the limit, \(\lim_{x \to 0^+} \sqrt{x}\), is indeed 0. This understanding of behavior greatly aids in determining limits and demonstrates why this particular limit does not exist from the left of zero but does from the right.