Finding the limit of this expression

In summary, when x approaches 1 from the positive side, the denominator gets closer to 0, which makes the limit positive infinity. When x approaches 1 from the negative side, the limit becomes negative infinity.
  • #1
turutk
15
0

Homework Statement



The limit of [tex]\frac{8x^3+x^2-5x}{3x^4-5x^2+2}[/tex] as x goes to 1

Homework Equations



Evaluate this limit algebraically.

The Attempt at a Solution



Tried using 1 instead of x. The denominator becomes 0.
Divided denominator by (x-1) but couldn't continue.
 
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  • #2
When x is near 1, the numerator will be near 4, which is a positive number.

When x is near 1, but larger than 1, the denominator is near zero and is positive. When x is near 1, but less than 1, the denominator is near zero and is negative.

Can you conclude anything from this information, about the limit of your rational function as x approaches 1?
 
  • #3
Mark44 said:
When x is near 1, the numerator will be near 4, which is a positive number.

When x is near 1, but larger than 1, the denominator is near zero and is positive. When x is near 1, but less than 1, the denominator is near zero and is negative.

Can you conclude anything from this information, about the limit of your rational function as x approaches 1?

does that mean limit does not exist?
 
  • #4
Yes, but can you clarify how you reached that conclusion?
 
  • #5
Mark44 said:
Yes, but can you clarify how you reached that conclusion?

at first thank for your help
i believe so.

while x is approaching 1 from + side the denominator gets closer to 0 which makes the limit positive infinity as x approaches 1+

while x is approaching 1 from - side, the limit becomes negative infinity.

one side of x=1 goes to -infinity the other +infinity so i conclude that the limit does not exist.

now i am working on this limit:

greatest integer(sinx) / greatest integer(x)
x approaches to 0

i predict that result will be 0 but i am not sure
 
  • #6
[tex]\left\lfloor[/tex]
turutk said:
at first thank for your help
i believe so.

while x is approaching 1 from + side the denominator gets closer to 0 which makes the limit positive infinity as x approaches 1+

while x is approaching 1 from - side, the limit becomes negative infinity.

one side of x=1 goes to -infinity the other +infinity so i conclude that the limit does not exist.
OK, good. Since the left-side limit is different from the right-side limit, the limit itself doesn't exist.
turutk said:
now i am working on this limit:

greatest integer(sinx) / greatest integer(x)
x approaches to 0

i predict that result will be 0 but i am not sure
I don't think it's zero, but I'm not sure, either. I would make a table of values of x and sin(x) values for x near zero on either side. Keep in mind that when -pi/2 < sin(x) < 0,
[tex]\left \lfloor{sin(x)}\right \rfloor = -1[/tex]
 
  • #7
Mark44 said:
[tex]\left\lfloor[/tex]OK, good. Since the left-side limit is different from the right-side limit, the limit itself doesn't exist.

I don't think it's zero, but I'm not sure, either. I would make a table of values of x and sin(x) values for x near zero on either side. Keep in mind that when -pi/2 < sin(x) < 0,
[tex]\left \lfloor{sin(x)}\right \rfloor = -1[/tex]

turns out it is zero:
http://www.wolframalpha.com/input/?...nx))/greatest+integer(x)+as+x+approaches+to+2

i still cannot explain
 

Related to Finding the limit of this expression

1. What is the purpose of finding the limit of an expression?

The limit of an expression represents the value that the expression approaches as the independent variable approaches a specific value. It helps us understand the behavior of the expression and its output as the input value gets closer and closer to the specified value.

2. How do I find the limit of an expression?

To find the limit of an expression, you can use algebraic techniques such as factoring, simplifying, and expanding the expression. You can also use graphical methods by plotting the graph of the expression and observing its behavior as the input value gets closer to the specified value.

3. What are the common methods used to evaluate limits?

The most commonly used methods to evaluate limits are the substitution method, the algebraic method, and the graphical method. The substitution method involves substituting the input value into the expression and evaluating it. The algebraic method involves using algebraic techniques to simplify the expression. The graphical method involves plotting the graph of the expression and observing its behavior as the input value gets closer and closer to the specified value.

4. Can limits be evaluated at any value?

No, limits can only be evaluated at values where the expression is defined. If the expression has a discontinuity at the specified value, the limit does not exist.

5. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the expression as the input value approaches the specified value from one direction, either from the left or the right. A two-sided limit considers the behavior of the expression from both directions, and the limit only exists if the left-hand limit and the right-hand limit are equal.

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