Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Find the limit of (a) \(\sqrt{f(x)}\), (b) \([3 f(x)]\), and (c) \([f(x)]^{2}\), as \(x\) approaches \(c\). $$ \lim _{x \rightarrow c} f(x)=9 $$

Step-by-Step Solution

Answer

The limits of the functions as \(x\) approaches \(c\) are: for \(\sqrt{f(x)}\) the limit is \(3\), for \(3f(x)\) the limit is \(27\), and for \((f(x))^2\) the limit is \(81\).

Step 1: Limit of the square root function

The limit of the function \(\sqrt{f(x)}\) as \(x\) approaches \(c\) is calculated by substituting the limit of \(f(x)\) as \(x\) approaches \(c\) into the function. That is, \(\sqrt{9}\), which equals \(3\). So the limit of the function \(\sqrt{f(x)}\) as \(x\) approaches \(c\) is \(3\).

Step 2: Limit of the function multiplied by 3

The limit of the function \(3f(x)\) as \(x\) approaches \(c\) is calculated by substituting the limit of \(f(x)\) as \(x\) approaches \(c\) into the function. That is, \(3*9\), which equals \(27\). Hence the limit of \(3f(x)\) as \(x\) approaches \(c\) is \(27\).

Step 3: Limit of the function squared

The limit of the function \((f(x))^2\) as \(x\) approaches \(c\) is found by substituting the limit of \(f(x)\) as \(x\) approaches \(c\) into the function. That is, \(9^2\), which equals \(81\). So, the limit of \((f(x))^2\) as \(x\) approaches \(c\) is \(81\).

Key Concepts

Limit of a Function

Understanding the concept of a limit is fundamental in calculus. A limit describes the value that a function approaches as the input (or argument) of the function approaches some value. For example, if we're told that \[\lim _{x \rightarrow c} f(x)=9\], we interpret this to mean that as we get closer and closer to some number 'c' with our 'x', the function 'f(x)' gets closer and closer to the number 9. This does not necessarily mean that 'f(x)' will equal 9 when 'x' is equal to 'c', just that it is the trend as 'x' approaches 'c'. It's like saying, no matter how close we get to 'c', 'f(x)' is nearly 9.

When solving limit problems, there are systematic strategies to follow. These often involve substituting 'c' into the function, if the function is continuous at that point, or employing other limit properties and theorems when it's not.

Square Root Function Limit

The limit of a square root function as 'x' approaches a particular value can be found using a similar strategy we use for other functions. Assuming we are given \[\lim _{x \rightarrow c} f(x)=9\] and we need to find the limit of \(\sqrt{f(x)}\), as 'x' approaches 'c', we'll utilize the value we know 'f(x)' is approaching. If 'f(x)' is approaching 9, then the square root of 'f(x)' will approach the square root of 9, which is 3. Therefore, \[\lim _{x \rightarrow c} \sqrt{f(x)}=3\].

This calculation hinges on the continuity of the square root function at the point where 'f(x)' equals 9. Understanding when and why such a function is continuous can help you better grasp the subtleties of limits involving square roots.

Function Multiplication Limit

When you're dealing with limits involving multiplication by a constant, you can simplify the process by applying the limit to the function first and then multiplying the result by the constant. According to the problem given, we can find \[\lim _{x \rightarrow c} [3 f(x)]\] by multiplying the constant 3 by the limit value of 'f(x)' as 'x' approaches 'c', which is known to be 9. Hence, we simply calculate 3 times 9 to get 27. Therefore, \[\lim _{x \rightarrow c} [3 f(x)]=27\].

Remember that multiplication by a constant is a linear operation, and linear operations are 'friendly' when it comes to limits—they can be moved in and out of the limit operator without altering the final result. This property greatly simplifies calculations and is widely used in more complex limit problems.

Function Squaring Limit

To deal with limits where the function is raised to a power, such as squaring, we first compute the limit of the function itself and then raise the resulting limit to the appropriate power. For the given problem, we know that \[\lim _{x \rightarrow c} f(x)=9\], so to find \[\lim _{x \rightarrow c} [f(x)]^{2}\], we simply square the known limit, resulting in 81. Therefore, \[\lim _{x \rightarrow c} [f(x)]^{2}=81\].

This is a direct application of a key limit law known as the power rule for limits. It works seamlessly for positive integer powers and, with some additional considerations, can also apply to other types of exponents. Recognizing and applying limit laws effectively is an essential skill in calculus and can streamline the process of calculating complex limits.