Limit of Rational Function....3

In summary, nycmathdad has provided a detailed explanation of the limit as x approaches −3 from the left side.
  • #1
nycmathdad
74
0
Find the limit of 1/(x^2 - 9) as x tends to -3 from the left side.

Approaching -3 from the left means that the values of x must be slightly less than -3.

I created a table for x and f(x).

x...(-4.5)...(-4)...(-3.5)
f(x)... 0.088...0.142...…...0.3076

I can see that f(x) is getting larger and larger and possibly without bound.

I say the limit is positive infinity.

Yes?
 
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  • #2
Problem 1.5.33.
Odd numbered.
Look up the answer.
 
  • #3
For x close to 3 and less than 3, the denominator, x^2- 9, is close to 0 and negative while the numerator, 1, is positive. Therefore?
 
  • #4
Country Boy said:
For x close to 3 and less than 3, the denominator, x^2- 9, is close to 0 and negative while the numerator, 1, is positive. Therefore?

Thus, the limit is positive infinity.
 
  • #5
Because the value is negative the limit is positive? That is what you are saying!

If x= 2.9, x^2- 9= -0.59, 1/(x^2- 9)= -1.69491525.
If x= 2.99, x^2- 9= -0.0599. 1/(x^2- 9)= -16.694408.
If x= 2.999, x^2- 9= -166.944.

That is NOT going to be positive!
 
  • #6
Beer soaked ramblings follow.
Country Boy said:
nycmathdad said:
Find the limit of 1/(x^2 - 9) as x tends to -3 from the left side.

Approaching -3 from the left means that the values of x must be slightly less than -3.

I created a table for x and f(x).

x...(-4.5)...(-4)...(-3.5)
f(x)... 0.088...0.142...…...0.3076

I can see that f(x) is getting larger and larger and possibly without bound.

I say the limit is positive infinity.

Yes?
Because the value is negative the limit is positive? That is what you are saying!

If x= 2.9, x^2- 9= -0.59, 1/(x^2- 9)= -1.69491525.
If x= 2.99, x^2- 9= -0.0599. 1/(x^2- 9)= -16.694408.
If x= 2.999, x^2- 9= -166.944.

That is NOT going to be positive!
Country Boy, in case you haven't noticed it yet, nycmathdad has been banned already.
Also, nycmathdad's conjecture about the function's behavior as x approaches −3 from the left (not +3 from the leftt as you seem to have misread), that is about the ratio 1/(x^2 − 9) becoming unbounded in the positive direction is indeed correct.
https://www.desmos.com/calculator/kh87f1cusv
 
Last edited:

What is a rational function?

A rational function is a function that can be expressed as a ratio of two polynomial functions. It can be written in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions.

What is the limit of a rational function?

The limit of a rational function is the value that the function approaches as the input (x) approaches a certain value. It can be found by evaluating the function at that value or by using algebraic techniques.

How do you find the limit of a rational function?

To find the limit of a rational function, you can first try to evaluate the function at the given value. If the function is undefined at that value, you can use algebraic techniques such as factoring, simplifying, or rationalizing the denominator to find the limit.

What is the difference between a finite and infinite limit of a rational function?

A finite limit of a rational function means that the function approaches a specific value as the input approaches a certain value. An infinite limit means that the function either approaches positive or negative infinity as the input approaches a certain value.

Why is it important to understand the limit of a rational function?

Understanding the limit of a rational function can help us determine the behavior of the function at certain points and can also be used to solve real-world problems in fields such as physics, economics, and engineering.

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