Mastering Limits In Calculus: Evaluating (1 – Cos(X))/X As X Approaches 0

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.

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