Determine whether the following statements are true and give an explanation or counterexample.


d. $\underset{x\to 0}{\lim }\sqrt{x}$ . (Hint: Graph y=√x)

Verified step by step guidance

1. Consider the function \( y = \sqrt{x} \) and its behavior as \( x \to 0 \).
2. Note that the square root function \( \sqrt{x} \) is only defined for \( x \geq 0 \).
3. As \( x \to 0^+ \) (approaching 0 from the right), \( \sqrt{x} \to \sqrt{0} = 0 \).
4. Since \( \sqrt{x} \) is not defined for \( x < 0 \), the limit from the left does not exist.
5. Therefore, the limit \( \lim_{x \to 0} \sqrt{x} \) does not exist because it is only defined from the right.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. For example, the limit of √x as x approaches 0 examines the value that √x approaches as x gets closer to 0.

Square Root Function

The square root function, denoted as √x, is defined for non-negative values of x and returns the non-negative value whose square is x. This function is continuous and increases as x increases. Understanding its graph is crucial for analyzing limits, especially as it approaches 0, where the function's behavior can be visually interpreted.

Graphical Interpretation

Graphical interpretation involves analyzing the visual representation of a function to understand its behavior. For the function y = √x, the graph is a curve that starts at the origin (0,0) and rises to the right. By examining this graph, one can intuitively grasp how the function behaves as x approaches 0, which aids in determining the limit.