Problem 47
Question
CAS Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} \sqrt{x} $$
Step-by-Step Solution
Verified Answer
The limit is 0.
1Step 1: Understand the Problem
We need to find \( \lim_{x \rightarrow 0} \sqrt{x} \). This asks for the limit of the square root of \( x \) as \( x \) approaches zero.
2Step 2: Analyze the Expression for Non-Negative x
The function \( \sqrt{x} \) is defined for \( x \geq 0 \). As \( x \) approaches 0 from the positive side (that is, \( x \to 0^+ \)), \( \sqrt{x} \) approaches \( \sqrt{0} = 0 \).
3Step 3: Consider x Approaching 0 from Negative Side
For \( x < 0 \), \( \sqrt{x} \) is not defined in the real number system. Therefore, the limit from the left side \( (x \to 0^-) \) does not exist in the reals.
4Step 4: Determine Overall Limit
Since \( \sqrt{x} \) is undefined for \( x < 0 \), we only consider the limit from the positive side, which is \( 0 \). Thus, the overall limit exists and equals 0.
Key Concepts
Approaching LimitsSquare Root FunctionOne-Sided Limits
Approaching Limits
Approaching limits is a fundamental concept in calculus. It involves understanding what happens to a function as the input value gets closer and closer to a certain point. For instance, consider the expression \( \lim_{x \rightarrow a} f(x) \). This represents what value \( f(x) \) is approaching as \( x \) gets very near to \( a \).
An important aspect to remember about limits is that they describe behavior rather than direct computation. A function does not need to be defined at a point to have a limit there; it's all about approaching that point.
An important aspect to remember about limits is that they describe behavior rather than direct computation. A function does not need to be defined at a point to have a limit there; it's all about approaching that point.
- If both the right-hand and left-hand limits approach the same value, then the limit exists.
- If these one-sided limits approach different values, then the limit does not exist.
- Sometimes, by examining a graph or by substitution, you can determine whether a limit exists.
Square Root Function
The square root function, represented as \( \sqrt{x} \), is another essential concept in calculus and mathematics as a whole. This function only accepts non-negative inputs because square roots of negative numbers are not real numbers. When dealing with limits, especially, it is crucial to consider the domain of the function.
For the square root function, especially as \( x \to 0 \), you need to keep in mind that:
For the square root function, especially as \( x \to 0 \), you need to keep in mind that:
- \( \sqrt{x} \) is smooth and continuous for all \( x \geq 0 \).
- For \( x < 0 \), \( \sqrt{x} \) is not defined in the real number system.
- As \( x \) approaches zero from the right (positive numbers), \( \sqrt{x} \) approaches zero.
- Understanding the behavior of \( \sqrt{x} \) near 0 helps decipher the nature of limits involving this function.
One-Sided Limits
One-sided limits are special limits where we only consider the behavior of a function as it approaches a particular point from one direction. This concept is essential when dealing with functions that might not be defined in one direction.
Consider the limit \( \lim_{x \to a^-} f(x) \), which describes the limit of \( f(x) \) as \( x \) approaches \( a \) from the left (negative direction). Similarly, \( \lim_{x \to a^+} f(x) \) examines the behavior as \( x \) approaches \( a \) from the right (positive direction).
Consider the limit \( \lim_{x \to a^-} f(x) \), which describes the limit of \( f(x) \) as \( x \) approaches \( a \) from the left (negative direction). Similarly, \( \lim_{x \to a^+} f(x) \) examines the behavior as \( x \) approaches \( a \) from the right (positive direction).
- If a function is only defined in one direction, like \( \sqrt{x} \) for \( x \geq 0 \), we must use one-sided limits.
- One-sided limits are particularly useful when the function behaves differently on each side of a point.
- In many cases, to establish a two-sided limit's existence, the one-sided limits from both directions must agree.
Other exercises in this chapter
Problem 46
Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(F(x)=\frac{3}{9-x^{2}}\)
View solution Problem 47
In Problems 41-52, verify that the given equations are identities. \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\)
View solution Problem 47
Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or no
View solution Problem 47
Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 0^{-}} \frac{x}{|x|}$$
View solution