Exercises10. Let $f(x)=\frac{1}{\sqrt{x-2}}$.(a) Find The Domain Of $f$. Write The Answer In Interval Notation.(b) Find And Simplify $\frac{f(x+h)-f(x)}{h}$.

Exercises: Domain and Limit of a Function

1. Domain of a Function

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x that can be plugged into the function without causing any problems.

1.1 Finding the Domain of a Function

To find the domain of a function, we need to identify any restrictions on the input values. In the case of the function $f(x)=\frac{1}{\sqrt{x-2}}$, we need to consider the restrictions imposed by the square root and the denominator.

The square root of a number is defined only if the number is non-negative. Therefore, we must have $x-2 \geq 0$, which implies that $x \geq 2$.

On the other hand, the denominator of a fraction cannot be zero. Therefore, we must have $\sqrt{x-2} \neq 0$, which implies that $x-2 > 0$, or $x > 2$.

Combining these two restrictions, we find that the domain of the function is $x \geq 2$.

1.2 Writing the Domain in Interval Notation

To write the domain in interval notation, we use the following notation:

[a, b]$ represents the closed interval from a to b, inclusive. $[a, b)$ represents the half-open interval from a to b, inclusive on the left and exclusive on the right. $(a, b]$ represents the half-open interval from a to b, exclusive on the left and inclusive on the right. $(a, b)$ represents the open interval from a to b, exclusive on both ends. In this case, the domain of the function is $x \geq 2$, which can be written in interval notation as $[2, \infty)$. **2. Limit of a Function** The limit of a function is a fundamental concept in calculus that describes the behavior of a function as the input values approach a certain point. **2.1 Finding the Limit of a Function** To find the limit of a function, we can use various techniques, such as direct substitution, factoring, and canceling. In this case, we are asked to find the limit of the function $\frac{f(x+h)-f(x)}{h}$. To do this, we first need to find the expressions for $f(x+h)$ and $f(x)$. Using the definition of the function, we have: $f(x+h) = \frac{1}{\sqrt{(x+h)-2}}

$$f(x) = \frac{1}{\sqrt{x-2}}$$

Now, we can substitute these expressions into the limit:

$$\lim_{h \to 0} \frac{\frac{1}{\sqrt{(x+h)-2}} - \frac{1}{\sqrt{x-2}}}{h}$$

2.2 Simplifying the Limit

To simplify the limit, we can start by finding a common denominator for the two fractions:

$$\lim_{h \to 0} \frac{\frac{1}{\sqrt{(x+h)-2}} - \frac{1}{\sqrt{x-2}}}{h} = \lim_{h \to 0} \frac{\frac{\sqrt{x-2} - \sqrt{(x+h)-2}}{\sqrt{(x+h)-2}\sqrt{x-2}}}{h}$$

Now, we can simplify the expression further by canceling out the common factors:

$$\lim_{h \to 0} \frac{\frac{\sqrt{x-2} - \sqrt{(x+h)-2}}{\sqrt{(x+h)-2}\sqrt{x-2}}}{h} = \lim_{h \to 0} \frac{\sqrt{x-2} - \sqrt{(x+h)-2}}{h\sqrt{(x+h)-2}\sqrt{x-2}}$$

2.3 Evaluating the Limit

To evaluate the limit, we can use the fact that the limit of a difference quotient is equal to the derivative of the function.

In this case, we have:

$$\lim_{h \to 0} \frac{\sqrt{x-2} - \sqrt{(x+h)-2}}{h\sqrt{(x+h)-2}\sqrt{x-2}} = \frac{d}{dx} \left(\frac{1}{\sqrt{x-2}}\right)$$

Using the chain rule, we can find the derivative:

$$\frac{d}{dx} \left(\frac{1}{\sqrt{x-2}}\right) = -\frac{1}{2(x-2)^{3/2}}$$

Therefore, the limit of the function is:

\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = -\frac{1}{2(x-2)^{3/2}}$<br/> **Q&A: Domain and Limit of a Function** **Q: What is the domain of the function $f(x)=\frac{1}{\sqrt{x-2}}$?** A: The domain of the function is $x \geq 2$, which can be written in interval notation as $[2, \infty)$. **Q: Why is the domain restricted to $x \geq 2$?** A: The domain is restricted to $x \geq 2$ because the square root of a number is defined only if the number is non-negative. Therefore, we must have $x-2 \geq 0$, which implies that $x \geq 2$. **Q: What is the limit of the function $\frac{f(x+h)-f(x)}{h}$?** A: The limit of the function $\frac{f(x+h)-f(x)}{h}$ is $-\frac{1}{2(x-2)^{3/2}}$. **Q: How do you find the limit of a function?** A: To find the limit of a function, you can use various techniques, such as direct substitution, factoring, and canceling. In this case, we used the definition of the function and the fact that the limit of a difference quotient is equal to the derivative of the function. **Q: What is the derivative of the function $f(x)=\frac{1}{\sqrt{x-2}}$?** A: The derivative of the function $f(x)=\frac{1}{\sqrt{x-2}}$ is $-\frac{1}{2(x-2)^{3/2}}$. **Q: How do you use the chain rule to find the derivative of a function?** A: To use the chain rule to find the derivative of a function, you need to identify the outer and inner functions. In this case, the outer function is $\frac{1}{\sqrt{x-2}}$ and the inner function is $x-2$. You then apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. **Q: What is the significance of the limit of a function?** A: The limit of a function is a fundamental concept in calculus that describes the behavior of a function as the input values approach a certain point. It is used to define the derivative of a function and to study the properties of functions. **Q: How do you use the limit of a function to study the properties of functions?** A: You can use the limit of a function to study the properties of functions by analyzing the behavior of the function as the input values approach a certain point. This can help you understand the behavior of the function over a wider range of input values. **Q: What are some common applications of the limit of a function?** A: Some common applications of the limit of a function include: * Studying the behavior of functions over a wider range of input values * Defining the derivative of a function * Analyzing the properties of functions, such as continuity and differentiability * Solving optimization problems, such as finding the maximum or minimum value of a function **Q: How do you use the limit of a function to solve optimization problems?** A: You can use the limit of a function to solve optimization problems by analyzing the behavior of the function as the input values approach a certain point. This can help you find the maximum or minimum value of the function. **Q: What are some common mistakes to avoid when working with the limit of a function?** A: Some common mistakes to avoid when working with the limit of a function include: * Not checking for restrictions on the input values * Not using the correct techniques for finding the limit * Not checking for the existence of the limit * Not using the correct notation for the limit **Q: How do you check for the existence of the limit of a function?** A: You can check for the existence of the limit of a function by analyzing the behavior of the function as the input values approach a certain point. This can help you determine whether the limit exists or not. **Q: What are some common techniques for finding the limit of a function?** A: Some common techniques for finding the limit of a function include: * Direct substitution * Factoring * Canceling * Using the definition of the function * Using the chain rule **Q: How do you use the definition of a function to find the limit of a function?** A: You can use the definition of a function to find the limit of a function by analyzing the behavior of the function as the input values approach a certain point. This can help you determine the limit of the function. **Q: What are some common applications of the definition of a function?** A: Some common applications of the definition of a function include: * Defining the limit of a function * Defining the derivative of a function * Analyzing the properties of functions, such as continuity and differentiability * Solving optimization problems, such as finding the maximum or minimum value of a function