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

  • MHB
  • Thread starter karush
  • Start date
  • Tags
    Limit Zero
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
Gold Member
MHB
3,269
5
$$\lim_{{x}\to{0}}\frac{\cos\left({3x}\right)-1}{{x}^{2}}$$
$$\frac{f'}{g'}
=-\frac{3\sin\left({3x}\right)}{2x}
=-\frac{9}{2}\cdot\frac{\sin\left({3x}\right)}{3x}$$
$x\to 0$ is $-\frac{9}{2}$

Just seeing if this is correct or better way to do it
 
Physics news on Phys.org
  • #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.

Similar threads

MHB Limit $\frac{f(x)}{g(x)}$: Solve w/ L'H Rule
Replies
4
Views
2K
I Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##
Replies
1
Views
2K
MHB What is the Limit of (3x)/(x-2) as x Approaches 2 from the Left?
Replies
5
Views
1K
MHB Calculating $\displaystyle \lim_{x\to 0}$ Complex Limit
Replies
4
Views
1K
I Integration of √(1-x^2) with x=sinθ: Check Correctness and Simplification
Replies
12
Views
2K
B Having trouble deriving the volume of an elliptical ring toroid
Replies
4
Views
2K
MHB What is the Slant Asymptote of $\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$?
Replies
3
Views
1K
I Question about Limit: \lim_{x\rightarrow1} (x+1)
Replies
29
Views
2K
MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$
Replies
1
Views
2K
I Why is the integral of ##\arcsin(\sin(x))## so divisive?
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top