limit as x approaches 0: 1-cosx/x
0
To evaluate the limit as x approaches 0 of (1 – cos(x))/x, we can try to directly substitute 0 for x, but this yields an indeterminate form of 0/0. So, we can use L’Hopital’s rule or algebraic manipulation to simplify the expression.
Using L’Hopital’s rule, we can take the derivative of the numerator and denominator separately with respect to x:
lim x→0 (1 – cos(x))/x = lim x→0 sin(x)/1 = sin(0)/1 = 0/1 = 0
Alternatively, we can use the identity 1 – cos(x) = 2sin^2(x/2) to rewrite the expression as:
lim x→0 (1 – cos(x))/x = lim x→0 [2sin^2(x/2)]/[xsin(x/2)]
We can cancel out the sin(x/2) term in the denominator and simplify the expression further:
lim x→0 (1 – cos(x))/x = lim x→0 [2sin(x/2)]/x = lim x→0 [sin(x/2)/(x/2)] * (x/2) * 2 = 1 * 0 * 2 = 0
Therefore, the limit as x approaches 0 of (1 – cos(x))/x is equal to 0.
More Answers:
[next_post_link]