Problem 37
Question
Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 0^{+}} f(x)\) and \(\lim _{x \rightarrow 0^{-}} f(x)\), where $$ f(x)=\left\\{\begin{array}{ll} 2 x & \text { if } x<0 \\ x^{2} & \text { if } x \geq 0 \end{array}\right. $$
Step-by-Step Solution
Verified Answer
The one-sided limits of the given function are: \(\lim _{x \rightarrow 0^{-}} f(x) = 0\) and \(\lim _{x \rightarrow 0^{+}} f(x) = 0\).
1Step 1: Analyze function for x < 0
For values of x less than 0, the function is given by \(f(x) = 2x\). To find the limit of the function as x approaches 0 from the left, we can simply substitute the value of x in the expression of the function.
Step 2: Substitute value of x and find the limit
2Step 2: Calculate the limit as x approaches 0 from the left
The limit as \(x \rightarrow 0^{-}\) is given by
\[
\lim _{x \rightarrow 0^{-}} f(x) = \lim_{x \rightarrow 0^{-}} (2x) = 2(0) = 0
\]
Step 3: Evaluate the one-sided limit as x approaches 0 from the right (x ≥ 0):
3Step 3: Analyze function for x ≥ 0
For values of x greater than or equal to 0, the function is given by \(f(x) = x^2\). To find the limit of the function as x approaches 0 from the right, we can substitute the value of x in the expression of the function.
Step 4: Substitute value of x and find the limit
4Step 4: Calculate the limit as x approaches 0 from the right
The limit as \(x \rightarrow 0^{+}\) is given by
\[
\lim _{x \rightarrow 0^{+}} f(x) = \lim_{x \rightarrow 0^{+}} (x^2) = (0)^2 = 0
\]
Thus, the one-sided limits of the function are:
\[
\lim _{x \rightarrow 0^{-}} f(x) = 0
\]
\[
\lim _{x \rightarrow 0^{+}} f(x) = 0
\]
Key Concepts
Piecewise FunctionsLimit from the LeftLimit from the Right
Piecewise Functions
A piecewise function is a type of function composed of multiple sub-functions, each defined on a specific interval of the function's domain. These sub-functions are separated by specific conditions that dictate which part of the piecewise function to use based on the input value.
For example, the given function \(f(x)\) is a classic piecewise function because it has two distinct parts based on the condition of the input \(x\):
This means that as \(x\) transitions across different values, the rule by which it operates changes. Understanding where and why each sub-function applies is key to mastering piecewise functions.
For example, the given function \(f(x)\) is a classic piecewise function because it has two distinct parts based on the condition of the input \(x\):
- For \(x < 0\) (values to the left of zero), the function used is \(f(x) = 2x\).
- For \(x \geq 0\) (values equal to or greater than zero), the function used is \(f(x) = x^2\).
This means that as \(x\) transitions across different values, the rule by which it operates changes. Understanding where and why each sub-function applies is key to mastering piecewise functions.
Limit from the Left
The limit from the left (denoted as \(\lim_{x \to c^{-}} f(x)\)) refers to approaching a specific point \(c\) from values less than \(c\). In essence, this examines the behavior of a function as it gets closer to this point from the left side.
In the context of the problem, we wish to find \(\lim_{x \to 0^{-}} f(x)\) where the function has the form \(2x\) for values of \(x\) less than 0. Since we're interested in \(x < 0\), we use this part of the piecewise function to determine the behavior as \(x\) approaches zero from the negative side.
By substitution:
This means that as \(x\) approaches 0 from the left, the function output approaches 0.
In the context of the problem, we wish to find \(\lim_{x \to 0^{-}} f(x)\) where the function has the form \(2x\) for values of \(x\) less than 0. Since we're interested in \(x < 0\), we use this part of the piecewise function to determine the behavior as \(x\) approaches zero from the negative side.
By substitution:
- Calculate \(\lim_{x \to 0^{-}} 2x = 2(0) = 0\)
This means that as \(x\) approaches 0 from the left, the function output approaches 0.
Limit from the Right
Conversely, the limit from the right (denoted as \(\lim_{x \to c^{+}} f(x)\)) examines the behavior of the function as it nears a point \(c\) from values greater than \(c\). This is crucial for understanding how a function behaves as it approaches a certain point from larger values.
In our exercise problem, to find \(\lim_{x \to 0^{+}} f(x)\), we look at the part of the piecewise function applied for \(x \geq 0\), which is \(x^2\). Here, our interest is in the behavior as \(x\) tends towards zero from positive values.
By substitution:
This tells us that as \(x\) approaches zero from the right, the output of the function also approaches 0. Together, these results from both one-sided limits inform us that the function behaves consistently at \(x = 0\), with a final one-sided limit value of zero from both directions.
In our exercise problem, to find \(\lim_{x \to 0^{+}} f(x)\), we look at the part of the piecewise function applied for \(x \geq 0\), which is \(x^2\). Here, our interest is in the behavior as \(x\) tends towards zero from positive values.
By substitution:
- Calculate \(\lim_{x \to 0^{+}} x^2 = (0)^2 = 0\)
This tells us that as \(x\) approaches zero from the right, the output of the function also approaches 0. Together, these results from both one-sided limits inform us that the function behaves consistently at \(x = 0\), with a final one-sided limit value of zero from both directions.
Other exercises in this chapter
Problem 37
In Exercises 37-40, find each limit by evaluating the derivative of a suitable function at an appropriate point. Hint: Look at the definition of the derivative.
View solution Problem 37
The gross domestic product (GDP) of a certain country is projected to be $$ N(t)=t^{2}+2 t+50 \quad(0 \leq t \leq 5) $$ billion dollars \(t\) yr from now. What
View solution Problem 37
Find the indicated limit. \(\lim _{x \rightarrow-3} \sqrt{2 x^{4}+x^{2}}\)
View solution Problem 38
Find the derivative of the function. \(g(x)=\sqrt{\frac{2 x+1}{2 x-1}}\)
View solution