limit as x approaches 0: 1-cosx/x
To evaluate the limit as x approaches 0 for the expression 1 – cos(x)/x, we can use L’Hôpital’s Rule or some trigonometric properties
To evaluate the limit as x approaches 0 for the expression 1 – cos(x)/x, we can use L’Hôpital’s Rule or some trigonometric properties.
Let’s start by using a trigonometric identity: cos(x) = 1 – (sin(x))^2.
Therefore, the expression can be rewritten as:
1 – cos(x)/x = 1 – (1 – (sin(x))^2)/x
Now, we have a difference of squares in the numerator, so we can simplify further:
= 1 – (1 – sin(x))(1 + sin(x))/x
Next, we can factor out a -1 from the numerator:
= 1 – (-1 + sin(x))(1 + sin(x))/x
Rearranging the numerator, we have:
= 1 + (sin(x) – 1)(1 + sin(x))/x
Now, let’s simplify the expression further by multiplying the terms:
= 1 + (sin^2(x) – 1)/x
= 1 + sin^2(x)/x – 1/x
We can simplify the expression even more:
= sin^2(x)/x
Now, as x approaches 0, sin(x)/x approaches 1 (a well-known limit), so we have:
lim(x->0) sin^2(x)/x = lim(x->0) sin(x)/x * sin(x)
= 1 * 0
= 0
Therefore, the limit of 1 – cos(x)/x as x approaches 0 is equal to 0.
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