Let f(x)=(2)/(x) and g(x)=(3)/(x-1) . Find limx →1(f ∘g)(x) and limx →1(g ∘f)(x).

Name: Problem 53, Chapter 11: Precalculus
Uploaded: 2021-08-30T17:29:55-08:00
Duration: 2 min 13 s
Channel: Farnood Ensan
Description: Let f(x)=(2)/(x) and g(x)=(3)/(x-1) . Find limx →1(f ∘g)(x) and limx →1(g ∘f)(x).

step 1 So this is a definition of F and G. And given that, we need to find F of G of X, which is going

Let f(x)=(2)/(x) and g(x)=(3)/(x-1) . Find limx →1(f ∘g)(x)...

Key Concepts

Function Composition

Function composition involves applying one function to the results of another function. It is a fundamental operation in mathematics where, given two functions f and g, the composite function (f ? g)(x) is defined as f(g(x)). This concept is essential because the properties and limits of the composite function depend on the behaviors of both f and g, as well as on how the output of g fits within the domain of f.

Limit of a Function

The limit of a function describes its behavior as the input approaches a particular value. Limits are used to analyze the behavior of functions near points of interest, including points where the function may not be explicitly defined. Understanding limits is crucial for studying continuity, analyzing asymptotic behavior, and dealing with indeterminate forms.

Domain Considerations

When working with composite functions and limits, it is important to consider the domain of the functions involved. Some functions can have restrictions or discontinuities (such as points where division by zero occurs), which can affect the evaluation of a limit. Analyzing domain issues ensures that the composite function is defined near the point of interest and helps in understanding potential vertical asymptotes or discontinuities.

Algebraic Manipulation in Limits

Algebraic manipulation is often a key step in evaluating limits, especially for composite functions. Simplifying expressions can make it easier to substitute values or apply limit rules, and can also help in resolving indeterminate forms. This technique is critical in limit evaluations to reveal the underlying behavior of the function as the variable approaches a specific value.

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