Let f(x)=x^3-x^2+5 x-1 and g(x)=2 . Find limx → 3(f ∘ g)(x) and limx → 3(g ∘ f)(x)

Name: Problem 51, Chapter 11: Precalculus
Uploaded: 2020-03-19T16:30:59-08:00
Duration: 2 min 34 s
Channel: Elizabeth Xu
Description: Let f(x)=x^3-x^2+5 x-1 and g(x)=2 . Find limx → 3(f ∘ g)(x) and limx → 3(g ∘ f)(x)

step 1 Now we first want to find the limit as x approaches 3 of f composed with g of x. This is essenti

Let f(x)=x^3-x^2+5 x-1 and g(x)=2 . Find limx → 3(f...

$$
\begin{array}{l}{\text { Let } f(x)=x^{3}-x^{2}+5 x-1 \text { and } g(x)=2 . \text { Find } \lim _{x \rightarrow 3}(f \circ g)(x)} \\ {\text { and } \lim _{x \rightarrow 3}(g \circ f)(x)}\end{array}
$$

Key Concepts

Direct Substitution in Limits

Direct substitution is a method of evaluating limits where the limiting value is substituted directly into the function, provided the function is continuous at that point. This concept is particularly relevant when dealing with compositions and polynomial functions, where the continuity allows an immediate evaluation.

Constant Functions

A constant function is one that returns the same value regardless of the input. When a constant function is involved in composition or limit evaluation, it significantly simplifies the process because its output does not vary with the input, making the limit evaluation straightforward.

Limits

Limits are a fundamental concept in calculus and analysis, describing the behavior of a function as its input approaches a particular value. The evaluation of limits is key in understanding continuity, convergence, and the overall behavior of functions at specific points, especially when working with compositions of functions.

Function Composition

Function composition involves applying one function to the results of another. In the context of limits, it requires understanding how the output of the inner function becomes the input of the outer function. It is essential to use this concept when evaluating limits that involve compositions, as it may simplify the problem by reducing it to the evaluation of a single, often simpler, limit.

Transcript

Please give Ace some feedback