About Limit of (1-cos(x))/x as x approaches 0
The limit of (1-cos(x))/x as x approaches 0 is 0. Substituting x = 0 gives 0/0, so use L'Hopital's rule: the derivative of the top is sin(x), the derivative of the bottom is 1, and sin(0)/1 = 0. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The calculator below opens with this limit already worked, and the full steps are written out above. The method it reports is l'hopital's rule. Change the expression or the point to try your own, or switch the direction to see each side on its own.
On the graph, the dashed line marks where x is heading and the horizontal line marks the limit value, so you can see the curve close in on it.
How to use Limit of (1-cos(x))/x as x approaches 0
- Type the expression in x, for example (x^2-4)/(x-2), or use the key pad.
- In "x approaches" enter a number such as 2, a constant like pi/2, or inf or -inf.
- For a finite point pick Both sides, From the left or From the right. At infinity there is only one direction.
- Read the result card for the limit and the name of the method used.
- Open the steps to see the working, and use the graph to check the behaviour near the point.
- If the result is labelled a numeric estimate, treat it as a guide and look for an algebraic argument.
Features
- Limits at a number, at infinity and at minus infinity.
- Left-hand, right-hand and two-sided limits, with the two-sided limit reported as not existing when the sides disagree.
- Plus and minus infinity as results, for expressions that grow without bound.
- Steps that name the method: substitution, factor and cancel, standard limit, L'Hopital's rule or leading terms.
- A numeric cross-check, and a clear label when the answer is a numeric estimate.
- A graph around the point with the limit drawn as a horizontal line and the point marked.
- A live preview and a math key pad, so input is easy on a phone.
- Clear errors for input that cannot be read.
The result and the reason
Substituting x = 0 gives 0/0, so use L'Hopital's rule: the derivative of the top is sin(x), the derivative of the bottom is 1, and sin(0)/1 = 0.
limit of (1-cos(x))/x as x approaches 0 is 0Worked examples
Limit of (1-cos(x))/x as x approaches 0 gives 0
Method: l'hopital's rule.
Privacy: how it works
- The limit is worked out by code running in your browser tab. Your expressions are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- If direct substitution gives a number, you are done: that number is the limit for any continuous function.
- For 0/0, try factoring and cancelling before reaching for L'Hopital's rule.
- At infinity, divide the top and bottom by the highest power of x, or just compare the leading terms.
- Check the one-sided limits when the function has an absolute value, a hole or a vertical asymptote at the point.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your expressions are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your expressions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is the limit of (1-cos(x))/x as x approaches 0?
It is 0. Substituting x = 0 gives 0/0, so use L'Hopital's rule: the derivative of the top is sin(x), the derivative of the bottom is 1, and sin(0)/1 = 0.
Which method finds the limit of (1-cos(x))/x as x approaches 0?
The calculator uses l'hopital's rule. The numbered steps above show exactly where it is applied.
Why is this limit 0 when (1-cos(x))/x^2 is not?
The numerator 1 - cos(x) behaves like x^2/2 near 0. Dividing by x leaves roughly x/2, which goes to 0. Dividing by x^2 leaves 1/2.
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