Problem 1
Question
Use the results of this section to evaluate the given limit. $$ \lim _{x \rightarrow 2}(-5) $$
Step-by-Step Solution
Verified Answer
The limit is -5.
1Step 1: Understanding the Problem
We need to evaluate the limit \( \lim_{x \rightarrow 2}(-5) \). This means we want to find the value that the expression approaches as \( x \) gets closer to 2.
2Step 2: Identifying Constant Limit Property
One important property of limits is that the limit of a constant function is the constant itself, regardless of the value of \( x \). Here, our function is a constant \(-5\).
3Step 3: Applying the Constant Limit Property
According to the constant limit property, \( \lim_{x \rightarrow a} c = c \) for any constant \( c \) and any value \( a \). Therefore, \( \lim_{x \rightarrow 2}(-5) = -5 \).
Key Concepts
Limits of FunctionsEvaluating LimitsLimit Properties
Limits of Functions
Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a particular point. Understanding limits helps in analyzing the precise behavior of functions that might not be explicitly defined at that point.
For instance, if we examine the limit \( \lim_{x \to a} f(x) \), we're interested in knowing what value \( f(x) \) gets closer to as \( x \) approaches the value \( a \). It's important to remember that limits aren't necessarily about what happens at \( x = a \); rather, they're about the trend of \( f(x) \) as \( x \) is close to \( a \).
For instance, if we examine the limit \( \lim_{x \to a} f(x) \), we're interested in knowing what value \( f(x) \) gets closer to as \( x \) approaches the value \( a \). It's important to remember that limits aren't necessarily about what happens at \( x = a \); rather, they're about the trend of \( f(x) \) as \( x \) is close to \( a \).
- Limits help in understanding the continuity and discontinuity of functions.
- They are used to find instantaneous rates of change, which are central in the study of derivatives.
- The concept of limits also lays the groundwork for defining the concept of continuity and integrals.
Evaluating Limits
Evaluating limits involves finding the value that a function approaches as the input approaches a certain point. There are several methods to evaluate limits:
1. **Direct Substitution**: This is the simplest method where you substitute the value directly into the function, provided it does not result in an indeterminate form such as \( \frac{0}{0} \).
2. **Factoring**: Useful for simplifying a function that is initially in an indeterminate form by factoring and canceling terms.
3. **Rationalization**: Often used when limits involve square roots, allowing us to simplify expressions before evaluating the limit.
In our example, the function you're working with is constant: \( -5 \).
Hence, regardless of the value of \( x \), as \( x \) gets closer to 2, the function will remain \( -5 \). Thus, by merely recognizing this property, you can evaluate this specific limit easily without more sophisticated calculus techniques.
1. **Direct Substitution**: This is the simplest method where you substitute the value directly into the function, provided it does not result in an indeterminate form such as \( \frac{0}{0} \).
2. **Factoring**: Useful for simplifying a function that is initially in an indeterminate form by factoring and canceling terms.
3. **Rationalization**: Often used when limits involve square roots, allowing us to simplify expressions before evaluating the limit.
In our example, the function you're working with is constant: \( -5 \).
Hence, regardless of the value of \( x \), as \( x \) gets closer to 2, the function will remain \( -5 \). Thus, by merely recognizing this property, you can evaluate this specific limit easily without more sophisticated calculus techniques.
Limit Properties
Exploring limit properties is essential for solving more complex limit problems. There are several important properties that aid in limit evaluations:
1. **Constant Rule**: As seen in the exercise, the limit of a constant \( c \) as \( x \) approaches any value \( a \) is the constant itself, i.e., \( \lim_{x \to a} c = c \).
2. **Sum and Difference Rule**: \( \lim_{x \to a} (f(x) \pm g(x)) = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \). This allows evaluating the limits of sums and differences by evaluating individual limits separately.
3. **Product Rule**: \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \). This property is applied when functions are multiplied together.
4. **Quotient Rule**: Provides that \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) given \( \lim_{x \to a} g(x) eq 0 \).
These properties simplify complex functions into manageable parts, making evaluation efficient. Recognizing and applying these properties empowers you to tackle a wide range of limit problems with confidence.
1. **Constant Rule**: As seen in the exercise, the limit of a constant \( c \) as \( x \) approaches any value \( a \) is the constant itself, i.e., \( \lim_{x \to a} c = c \).
2. **Sum and Difference Rule**: \( \lim_{x \to a} (f(x) \pm g(x)) = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \). This allows evaluating the limits of sums and differences by evaluating individual limits separately.
3. **Product Rule**: \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \). This property is applied when functions are multiplied together.
4. **Quotient Rule**: Provides that \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) given \( \lim_{x \to a} g(x) eq 0 \).
These properties simplify complex functions into manageable parts, making evaluation efficient. Recognizing and applying these properties empowers you to tackle a wide range of limit problems with confidence.
Other exercises in this chapter
Problem 1
Determine whether \(f\) is continuous at \(a\). $$ f(x)=\sqrt{x} \cos x ; a=\frac{\pi}{4} $$
View solution Problem 1
In Exercises \(1-8\) guess the value of the limit. \(\lim _{x \rightarrow-1}(x+4)\)
View solution Problem 2
Determine the one-sided limit. $$ \lim _{x \rightarrow \pi / 3^{-}} \cos (x+\pi / 6) $$
View solution Problem 2
Determine whether \(f\) is continuous at \(a\). $$ f(x)=e^{x} \ln x ; a=1 $$
View solution