- #1
blaze33
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Homework Statement
limit as x->1/3 of (2-6x)^2/(3x-1)(9x^2-1)
Homework Equations
The Attempt at a Solution
substituting gets 0/0, tried expanding but it doesn't work either, don't know what else is there left to do...
No, this should be 4(3x - 1)(3x -1).blaze33 said:simplifying the numerator gives 36x^2-24x+4
factor it to (12x+4)(3x+1)
factor it again to 4(3x+1)(3x+1)
blaze33 said:so
4(3x+1)(3x+1)/(3x-1)(9x^2-1)
can i cancel (3x+1) and (3x-1) ?
well i tried canceling it anyway and then substituted 1/3 again now i get 8/0 so its +infinity i guess.
Mark44 said:No, this should be 4(3x - 1)(3x -1).
It would have been simpler to recognize that (2 - 6x)^2 = (6x - 2)^2 = 4(3x - 1)^2.
Can you cancel 3 + 1 and 3 - 1?
physicsman2 said:actually i factored out the numerator a different way
i took out the 4 from the beginning and got 4(9x^2 - 6x + 1) then got (3x-1)(3x-1)
you neglected the fact that it -24x, not +24x when you factored it
for the denominator: (3x-1)(9x^2 - 1)
recognize the difference of squares in the denominator?
it will become(3x-1)(3x-1)(3x+1), so cancel and evaluate
just curious do you know about l'hopital's rule?
if you do, try to evaluate the limit using that rule and you see that the answer is different
just another example of when l'hopital's rule fails if you're interested in knowing that
physicsman2 said:did you see my post above? kinda pointed that out and helped you out a little more
Mark44 said:I hope you realized that when I asked "can you cancel 3 + 1 and 3 - 1" you understood me to mean that you can't do that. And you can't cancel 3x + 1 and 3x - 1.
So what did you get as your final expression before you took the limit?
A limit is a fundamental concept in mathematics that describes the behavior of a function as the input values approach a certain value. It is used to determine the value that a function approaches but may never actually reach.
Finding a limit is important because it allows us to understand the behavior of a function and make predictions about its values. It is also a crucial tool in calculus and other areas of mathematics.
To find a limit, you can use various methods such as plugging in values, using algebraic manipulation, or using the properties of limits. It is important to understand the rules and techniques for evaluating limits, as well as knowing when certain methods are appropriate.
The most common types of limits are one-sided limits, which involve approaching a value from either the left or right side, and two-sided limits, which involve approaching a value from both sides. Other types include infinite limits, where the function approaches positive or negative infinity, and limits at infinity, where the function approaches a specific value as the input grows larger or smaller.
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