Finding the limit of this expression

[turutk]

Homework Statement

The limit of \frac{8x^3+x^2-5x}{3x^4-5x^2+2} 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.

 

[Mark44]

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?

 

[turutk]

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?

 

[Mark44]

Yes, but can you clarify how you reached that conclusion?

 

[turutk]

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

 

[Mark44]

\left\lfloor

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.

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,
\left \lfloor{sin(x)}\right \rfloor = -1

 

[turutk]

\left\lfloorOK, 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,
\left \lfloor{sin(x)}\right \rfloor = -1

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

i still cannot explain

 

[Mark44]

No, this limit doesn't exist. When you used WolframAlpha, you had the limit as x approaches 2.

http://www.wolframalpha.com/input/?...nx))/greatest+integer(x)+as+x+approaches+to+0

 

[turutk]

No, this limit doesn't exist. When you used WolframAlpha, you had the limit as x approaches 2.

http://www.wolframalpha.com/input/?...nx))/greatest+integer(x)+as+x+approaches+to+0

now everything makes sense. thank you for your answers