Answer:
Answer:
There is a huge difference between evaluating a function and the limit of the function. Thus, these two concepts are not similar. When we evaluate a function, it's like an "input-output process". You have your input and you will have a corresponding output. While when we do limit of the function, technically we are not doing the input-out process but we are looking at where the function is going to as our x value is approaching a specific amount.
Different Types of Functions
Rational Function
Polynomial Function
Logarithmic Function
Trigonometric Function
The limit of a function is not the same as evaluating function. Let's take a look at a specific example.
Suppose you have the function, \begin{aligned}f(x)=\frac{x+1}{x^2-1}\end{aligned}
f(x)=
x
2
−1
x+1
.
We want to evaluate the function when x=1x=1 , so have to solve for f(1)f(1) .
\begin{gathered}\begin{aligned}f(x)&=\frac{x-1}{x^2-1}\\&=\frac{1-1}{1^2-1}\\&=\frac{0}{0}\end{aligned}\end{gathered}
f(x)
=
x
2
−1
x−1
=
1
2
−1
1−1
=
0
0
In this case, the value is undefined or it doesn't exist.
The idea here is that we are solving what is the function value when x is exactly equal to 1.
Now, let's take a look at the limit of the same function when xx approaches to 11 or \begin{aligned}\lim_{x \to 1}\frac{x+1}{x^2-1}\end{aligned}
x→1
lim
x
2
−1
x+1
.
\begin{gathered}\begin{aligned}\lim_{x \to 1}\frac{x-1}{x^2-1}&=\lim_{x \to 1}\frac{x-1}{(x+1)(x-1)}\\&=\lim_{x \to 1}\frac{1}{x+1}\\&=\lim_{x \to 1}\frac{1}{2}\\&=\frac{1}{2}\end{aligned}\end{gathered}
x→1
lim
x
2
−1
x−1
=
x→1
lim
(x+1)(x−1)
x−1
=
x→1
lim
x+1
1
=
x→1
lim
2
1
=
2
1
In this case, the answer is \begin{aligned}\frac{1}{2}\end{aligned}
2
1
.
The idea of doing the limit is that we are looking what is the direction of the function when x is approaching 1 but not exactly equal to 1.
Thus, these two concepts are totally different.
To learn more about the limit of the function, go to
Evaluating Function: https://brainly.ph/question/6112926
Evaluating Algebraic Expressions: https://brainly.ph/question/604877
Evaluating Circular Functions: https://brainly.ph/question/427575
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