Question

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{l} \text { - } \lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 10} f(x)=1, \quad f(1)=1 / 5 \\ \text { - } \lim _{x \rightarrow 1} g(x)=0, \quad \lim _{x \rightarrow 10} g(x)=\pi, \quad g(10)=\pi \end{array} $$ $$ \lim _{x \rightarrow 1} g(5 f(x)) $$

Short answer

The limit is \(\pi\).

Step-by-step solution

Identify the inner function limit

We need to find the limit of the inner function \(5f(x)\) as \(x\) approaches 1. Given that \(\lim _{x \rightarrow 1} f(x) = 2\), we can substitute this limit into the inner function. Therefore, \(\lim _{x \rightarrow 1} 5f(x) = 5 \times 2 = 10\).

Apply the limit to the outer function

With the limit of the inner function \(5f(x)\) as \(x\) approaches 1 being 10, we now need to evaluate the limit of \(g(10)\) as \(x\) approaches 1. From the given information, \(\lim _{x \rightarrow 10} g(x) = \pi\) and since \(x\) is approaching 1, we use the continuity of \(g(x)\) at 10.

Determine the final answer

We need \(\lim _{x \rightarrow 10} g(x)\) which results in \(\pi\). Therefore, \(\lim _{x \rightarrow 1} g(5f(x)) = \lim _{x \rightarrow 10} g(x) = \pi\).

Key concepts

Continuity

Continuity is one of the fundamental concepts in calculus that deal with how functions behave around a point. If we say a function is continuous at a point, it means there are no breaks, jumps, or holes at that point. Think of it like drawing a graph without lifting your pencil off the paper. This concept is crucial when dealing with limits, as continuity assures us that the limit of the function at a point is the same as the function’s value at that point.

For example, consider the function \(g(x)\) given in the exercise. At the point where \(x \rightarrow 10\), the function \(g(x)\) equals \(\pi\), which is exactly the limit value. This is a clear demonstration of continuity at that point. When a function is continuous like \(g(x)\) at \(x = 10\), it simplifies the process of evaluating limits because the values don’t suddenly leap to infinite or unknown values.

Function Composition

Function composition involves combining two functions to form a new function. The new function essentially applies one function to the results of another. It’s like taking two actions in sequence, where you first perform one task, and then use the outcome of that task in another one.

In the limit evaluation example, we dealt with a composed function \(g(5f(x))\). Here, \(5f(x)\) is calculated first, and then \(g\) is applied to the result. Such compositions can be tricky as each function can have its limits and behaviors, which affects the final result. Understanding how to manipulate composed functions is crucial when evaluating the limits because it involves figuring out how the change in one variable inside the inner function affects the output of the outer function.

Inner and Outer Functions

When analyzing function composition, it’s important to distinguish between the inner and outer functions. The inner function is the one that is computed first, and its result becomes the input for the outer function.

In our case, \(5f(x)\) serves as the inner function. We start by assessing the behavior of \(f(x)\) as \(x\) approaches 1, which we know approaches 2. Then, by multiplying with 5, we get \(5f(x) = 10\). The inner function simplifies to a direct number.

Then comes the outer function, \(g(x)\). The value from the inner function (10 in this instance) becomes the input for \(g(x)\). Understanding the role of each function helps in properly evaluating the final composite output, which in turn helps us determine the limit when \(x\) approaches a specific value.

Limit of a Function

The limit of a function helps in understanding the behavior of a function as the input approaches a specific point. It’s like predicting where the function is headed even if it never quite reaches that point.

In our exercise, we are tasked with evaluating \(\lim_{x \rightarrow 1} g(5f(x))\). By understanding and applying limits, we first consider what happens to \(f(x)\) as \(x\) approaches 1 – it’s simple since \(f(x)\) approaches 2. Substituting, \(5f(x)\) turns into 10. We then need \(g(10)\) as the final target of our limit evaluation.

The limit of the outer function based on the result of the inner function determines the overall limit of the combined composite function. Always check if the function remains continuous or if there are indicated behavior changes around those points to ensure accurate limit evaluations.