PBM.1 Limit to Zero: $$\lim_{x\to 0} \frac{\cos 3x-1}{x^2}$$

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In summary, the limit as x approaches 0 of (cos(3x)-1)/x^2 is equal to -9/2, which can be shown by using L'Hôpital's Rule or by making a substitution and simplifying the expression.
  • #1
karush
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$$\lim_{{x}\to{0}}\frac{\cos\left({3x}\right)-1}{{x}^{2}}$$
$$\frac{f'}{g'}
=-\frac{3\sin\left({3x}\right)}{2x}
=-\frac{9}{2}\cdot\frac{\sin\left({3x}\right)}{3x}$$
$x\to 0$ is $-\frac{9}{2}$

Just seeing if this is correct or better way to do it
 
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  • #2
That's correct, but if you are allowing yourself L'Hôpital, then I would just write:

\(\displaystyle L=\lim_{x\to0}\frac{\cos(3x)-1}{x^2}=-\frac{3}{2}\lim_{x\to0}\frac{\sin(3x)}{x}=-\frac{9}{2}\lim_{x\to0}\cos(3x)=-\frac{9}{2}\)
 
  • #3
Correct! Well done!
 
  • #4
\(\displaystyle \lim_{x\to0}\dfrac{\cos3x-1}{x^2}\)

\(\displaystyle x=\dfrac13y\)

\(\displaystyle \lim_{y\to0}\dfrac{9(\cos y-1)}{y^2}\)

\(\displaystyle \lim_{y\to0}\dfrac{-9\sin^2y}{y^2(\cos y+1)}=-9\cdot1\cdot\dfrac12=-\dfrac92\)
 
  • #5
Sorry but I don't see the purpose of sticking $y$ in this thing
 
  • #6
karush said:
Sorry but I don't see the purpose of sticking $y$ in this thing

Yes, you could do it without making a substitution, but it just looks cleaner if you do. However, the main takeaway from Greg's post is how to take the limit without using L'Hôpital's Rule at all. :)
 

FAQ: PBM.1 Limit to Zero: $$\lim_{x\to 0} \frac{\cos 3x-1}{x^2}$$

What is the meaning of PBM?

PBM stands for "problem-based learning", which is an approach to education that emphasizes solving real-world problems as a way to learn and develop critical thinking skills.

What does the limit to zero in the equation represent?

The limit to zero in this equation represents the behavior of the function as the input (x) approaches 0. It shows how the output (y) changes or approaches a specific value as x gets closer and closer to 0.

How do you calculate this limit?

This limit can be calculated by plugging in smaller and smaller values of x (approaching 0) into the equation and observing the resulting outputs. If the outputs get closer and closer to a specific value, then that value is the limit.

What is the significance of the number 3 in the equation?

The number 3 in the equation represents the coefficient of x in the cosine function, which determines the rate of change of the function. In this case, it affects how quickly the cosine function approaches 1 as x approaches 0.

How can the limit be used in real-world applications?

Limits can be used in many real-world applications, such as calculating the speed of an object as it approaches an obstacle, predicting the growth rate of a population, or analyzing the behavior of chemical reactions. It is a fundamental concept in calculus and helps us understand the behavior of functions and their inputs.

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