limit as x approaches 0: 1-cosx/x
To find the limit as x approaches 0 of the expression (1 – cosx)/x, we can use a trigonometric identity and apply algebraic manipulation
To find the limit as x approaches 0 of the expression (1 – cosx)/x, we can use a trigonometric identity and apply algebraic manipulation.
First, let’s rewrite the expression using the identity cos(0) = 1:
(1 – cosx)/x = (1 – cosx)/(x – 0)
Now, we can use the identity 2sin^2(x/2) = 1 – cosx:
(1 – cosx)/(x – 0) = (1 – cosx)/(2sin(x/2) * x/2)
Next, we can cancel out the common factor of cosx in the numerator and denominator:
(1 – cosx)/(2sin(x/2) * x/2) = 1/(2sin(x/2) * x/2)
Now, as x approaches 0, we can make the substitution x/2 = t:
1/(2sin(x/2) * x/2) = 1/(2sin(t) * t)
Finally, we can apply the limit:
lim x→0 (1 – cosx)/x = lim t→0 1/(2sin(t) * t)
To evaluate this limit, we can simplify the expression further by using the fact that sin(t)/t approaches 1 as t approaches 0:
lim t→0 1/(2sin(t) * t) = 1/(2 * 1 * 1) = 1/2
Therefore, the limit as x approaches 0 of (1 – cosx)/x is 1/2.
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