Calculating the Limit: $\lim_{x\to0}\frac{\sin x-x }{x^3}$

MHD93 · 2010-07-19T21:42:23+00:00

Hi Find the limit: \lim_{x\to0}\frac{\sin x-x }{x^3}

Thread starter MHD93
Start date Jul 19, 2010
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Limit

In summary, the limit of the function $\frac{\sin x-x}{x^3}$ as $x$ approaches $0$ is $-\frac{1}{6}$. To calculate the limit of a function, we substitute the value of $x$ into the function and observe the approaching value. The limit has significant applications in calculus, allowing us to understand the behavior of functions. A function can have a limit at a point where it is not defined. The limit can also help us determine the behavior of a function near a certain point by analyzing its approaching values.

Jul 19, 2010

MHD93

Hi

Find the limit:
[tex] \lim_{x\to0}\frac{\sin x-x }{x^3} [/tex]

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Jul 19, 2010

Mark44

Mentor

Insights Author

Mohammad_93 said:

Hi

Find the limit:
[tex] \lim_{x\to0}\frac{\sin x-x }{x^3} [/tex]

What have you tried?

Jul 19, 2010

MHD93

lim[x->0] (sinx - x)/x^3 = lim [x->0] sinx/x3 - 1/x2
but it's infinity - infinity!

Jul 19, 2010

Mark44

Mentor

Insights Author

So that didn't do you any good.

Do you know about L'Hopital's Rule?

Jul 20, 2010

hunt_mat

Homework Helper

Or the power series for sin x?

FAQ: Calculating the Limit: $\lim_{x\to0}\frac{\sin x-x }{x^3}$

1. What is the limit of the function $\frac{\sin x-x}{x^3}$ as $x$ approaches $0$?

The limit of the function $\frac{\sin x-x}{x^3}$ as $x$ approaches $0$ is $-\frac{1}{6}$. This means that as $x$ gets closer and closer to $0$, the value of the function approaches $-\frac{1}{6}$.

2. How do you calculate the limit of a function?

To calculate the limit of a function, you need to substitute the value that $x$ is approaching into the function and see what value the function approaches. If the function approaches a single value, that is the limit. If the function approaches different values from the left and right sides, the limit does not exist.

3. What is the significance of the limit in calculus?

The limit is a fundamental concept in calculus that allows us to analyze the behavior of functions and understand how they change over time or approach certain values. It is used to define important concepts such as continuity, differentiability, and the derivative.

4. Can a function have a limit at a point where it is not defined?

Yes, a function can have a limit at a point where it is not defined. This is because the limit only looks at the behavior of the function near that point, not necessarily at the point itself. As long as the function approaches a single value from both the left and right sides, the limit exists.

5. How can we use the limit to determine the behavior of a function near a certain point?

The limit can tell us whether a function is approaching a single value, approaching different values from the left and right sides, or approaching infinity or negative infinity. This information can help us determine the behavior of the function near that point and make predictions about its behavior at other points.