What Happens to f(x) as x Approaches 2?

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In summary, the conversation discussed investigating a limit of a function as x approaches a given value. The limit was found by evaluating the left-hand and right-hand limits, which were both equal to the function's value at the given value. The conclusion was that the limit was equal to this value, 4. The middle section of the piecewise function was also briefly mentioned, with the understanding that it did not affect the limit calculation.
  • #1
nycmathguy
Homework Statement
Determine a Limit Algebraically
Relevant Equations
Linear Expression
Quadratic Expression
Investigate A Limit

Investigate the limit of f(x) as x tends to c at the given c number.

Attachment has been deleted.

Let me see.

Let c = 2

I think I got to take the limit of f(x) as x tends to 2 from the left and right. What about as x tends to 2 (from the left and right at the same time)?

Find the limit of (x + 2) as x tends to 2 from the left side.

(2 + 2) = 4

Find the limit of x^2 as x tends to 2 from the right side.

(2)^2 = 4

LHL = RHL

Thus, the limit of f(x) as x tends to 2 is 4.

Is this right?

What about the middle section of this piecewise function? There we see f(x) is 4 if x = 2. I think we can say concerning the middle section that the limit of f(x) as x tends to 2 from the left and right at the same time is 4.

Yes?
 
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  • #2
If f(x) is continuous at x=c then approaches from the both sides coincide to f(c).
 
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  • #3
anuttarasammyak said:
If f(x) is continuous at x=c then approaches from the both sides coincide to f(c).

I did all this work for nothing, right?
 
  • #4
I just wrote the last conclusion. Effort to get it is an important issue.
 
  • #5
anuttarasammyak said:
I just wrote the last conclusion. Effort to get it is an important issue.

Yes, I try to show my part.

You said:

"If f(x) is continuous at x=c then approaches from the both sides coincide to f(c)."

Meaning? Is this the answer?
 
  • #6
nycmathguy said:
Homework Statement:: Determine A Limit Algebraically
Relevant Equations:: Linear Expression
Quadratic Expression

Investigate A Limit

Investigate the limit of f(x) as x tends to c at the given c number.

Attachment has been deleted.

Let me see.

Let c = 2

I think I got to take the limit of f(x) as x tends to 2 from the left and right. What about as x tends to 2 (from the left and right at the same time)?

Find the limit of (x + 2) as x tends to 2 from the left side.

(2 + 2) = 4

Find the limit of x^2 as x tends to 2 from the right side.

(2)^2 = 4

LHL = RHL

Thus, the limit of f(x) as x tends to 2 is 4.

Is this right?

What about the middle section of this piecewise function? There we see f(x) is 4 if x = 2. I think we can say concerning the middle section that the limit of f(x) as x tends to 2 from the left and right at the same time is 4.

Yes?
You've written a lot of stuff here, but much of it is extraneous (e.g., "let me see" etc., but you have omitted some important parts, such as the formula for the function (or an image of it).

What middle section are you asking about? Are there three piecewise definitions for this function, something like this:
##f(x) = \begin{cases}x + 2, & x < 2 \\
4, & x = 2 \\
x^2, & x > 2 \end{cases}##
 
  • #7
Mark44 said:
You've written a lot of stuff here, but much of it is extraneous (e.g., "let me see" etc., but you have omitted some important parts, such as the formula for the function (or an image of it).

What middle section are you asking about? Are there three piecewise definitions for this function, something like this:
$$f(x) = \begin {cases} x + 2 & x < 2 \\
4 & x = 2 \\
x^2 & x > 2 \end{cases}$$
f(x) = x + 2, for x

I deleted the picture by mistake a few days ago.
 
  • #8
Mark44 said:
You've written a lot of stuff here, but much of it is extraneous (e.g., "let me see" etc., but you have omitted some important parts, such as the formula for the function (or an image of it).

What middle section are you asking about? Are there three piecewise definitions for this function, something like this:
$$f(x) = \begin {cases} x + 2 & x < 2 \\
4 & x = 2 \\
x^2 & x > 2 \end{cases}$$
f(x) = x + 2, for x

Correct. This is the correct piecewise function.
Note: f(x) is 4 when x = 2 is what I call the middle part. I also call (x + 2) the top part and of course, x^2 is the bottom part. All three together make one function, a piecewise function.
 
  • #9
So if ##\lim_{x \to 2^-}f(x)## and ##\lim_{x \to 2^+}f(x)## both exist and are both equal to f(2), then ##\lim_{x \to 2}f(x)## exists and is equal to f(2) which is 4.

The notations ##2^-## and ##2^+## represent approaches from the left and right, respectively.
 
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  • #10
Mark44 said:
So if ##\lim_{x \to 2^-}f(x)## and ##\lim_{x \to 2^+}f(x)## both exist and are both equal to f(2), then ##\lim_{x \to 2}f(x)## exists and is equal to f(2) which is 4.

The notations ##2^-## and ##2^+## represent approaches from the left and right, respectively.

Very good. Didn't I conclude the limit is 4?
 
  • #11
In post #5 you weren't sure that you had answered the question.
 
  • #12
Mark44 said:
In post #5 you weren't sure that you had answered the question.

Mark,

I am receiving lots of warnings. Why? I am posting calculus questions in the calculus and beyond forum and precalculus in the precalculus forum. Why the warnings?
 
  • #13
nycmathguy said:
Mark,

I am receiving lots of warnings. Why? I am posting calculus questions in the calculus and beyond forum and precalculus in the precalculus forum. Why the warnings?
You currently have three warnings. One warning was for posting the URL for downloads of copyrighted books. The other two warnings were for post homework questions in forum sections other than the homework sections.
 
  • #14
Mark44 said:
You currently have three warnings. One warning was for posting the URL for downloads of copyrighted books. The other two warnings were for post homework questions in forum sections other than the homework sections.

I am starting out on the wrong track here.
 
  • #15
nycmathguy said:
What about the middle section of this piecewise function? There we see f(x) is 4 if x = 2. I think we can say concerning the middle section that the limit of f(x) as x tends to 2 from the left and right at the same time is 4.
The middle piece has nothing to do with the limit. You only care about what ##f## does as ##x## approaches 2, not about what ##f## actually does at ##x=2##. If f(2) was equal to, say, 0 or even if it was undefined, you would still conclude that the limit as ##x \to 2## exists and is equal to 4 because you showed both one-sided limits are equal to 4.

For the function you were given, since you have ##\lim_{x \to 2} f(x) = f(2)##, you could also conclude that ##f## is continuous at ##x=2##.
 
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  • #16
vela said:
The middle piece has nothing to do with the limit. You only care about what ##f## does as ##x## approaches 2, not about what ##f## actually does at ##x=2##. If f(2) was equal to, say, 0 or even if it was undefined, you would still conclude that the limit as ##x \to 2## exists and is equal to 4 because you showed both one-sided limits are equal to 4.

For the function you were given, since you have ##\lim_{x \to 2} = f(2)##, you could also conclude that ##f## is continuous at ##x=2##.
Good to know. As far as continuity is concerned, this concept is coming up a few sections later. I think I am posting too many questions. This is a very bad study habit. I will post no more than 3 problems per week per textbook.
 

Related to What Happens to f(x) as x Approaches 2?

1. What is a limit in mathematics?

A limit in mathematics is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a specific value.

2. Why is it important to investigate limits?

Investigating limits is important because it allows us to understand the behavior of a function and make predictions about its values. It also helps us to solve problems involving rates of change, continuity, and infinite series.

3. How do you investigate a limit?

To investigate a limit, you need to evaluate the function at values close to the limit value from both sides. If the values approach the same number, then the limit exists. You can also use algebraic techniques, such as factoring and simplifying, to evaluate limits.

4. What are the common types of limits?

The common types of limits include one-sided limits, where the function is evaluated from one side of the limit value, and two-sided limits, where the function is evaluated from both sides of the limit value. Other types include infinite limits, where the function approaches positive or negative infinity, and limits at infinity, where the function approaches a finite value as the input approaches infinity.

5. How do limits relate to derivatives and integrals?

Limits are closely related to derivatives and integrals. Derivatives are used to find the rate of change of a function at a specific point, which is essentially the slope of the tangent line at that point. Integrals, on the other hand, are used to find the area under a curve. Both derivatives and integrals involve the use of limits in their definitions and calculations.

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