lim x->0 1-cosx/x
0
To solve this limit, we can use L’Hôpital’s rule, which states that if the limit of f(x)/g(x) as x approaches some value results in a 0/0 or ∞/∞ indeterminate form, then we can take the derivative of both the numerator and denominator and re-evaluate the limit. We can keep applying L’Hôpital’s rule until the indeterminate form is resolved.
In this case, we have:
lim x->0 (1-cosx)/x
Since this is a 0/0 indeterminate form, we can use L’Hôpital’s rule and find the derivative of the numerator and denominator:
lim x->0 -sinx/1
The derivative of 1 is 0, so we are left with:
lim x->0 -sinx
Evaluating this limit, we get:
– sin(0) = 0
Therefore, the limit of (1-cosx)/x as x approaches 0 is 0.
In summary, the answer is:
lim x->0 (1-cosx)/x = 0
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