limit as x approaches 0: sinx/x
To evaluate the limit of sin(x)/x as x approaches 0, we can use the concept of L’Hôpital’s Rule
To evaluate the limit of sin(x)/x as x approaches 0, we can use the concept of L’Hôpital’s Rule.
L’Hôpital’s Rule states that if you have a limit of the form (f(x) / g(x)), where both f(x) and g(x) approach 0 or ±∞ as x approaches a certain value, and if the limit of the ratio of their derivatives, f'(x) / g'(x), exists, then the limit of the original function also exists and is equal to the limit of the ratio of the derivatives.
In this case, we have the limit sin(x)/x as x approaches 0. Both sin(x) and x approach 0 as x approaches 0, so we can apply L’Hôpital’s Rule.
Taking the derivatives of sin(x) and x, we have:
f(x) = sin(x), so f'(x) = cos(x)
g(x) = x, so g'(x) = 1
Now we can evaluate the limit of f'(x) / g'(x) as x approaches 0:
lim (x→0) (f'(x) / g'(x)) = lim (x→0) (cos(x) / 1) = cos(0) = 1
Since the limit of the ratio of derivatives is 1, according to L’Hôpital’s Rule, the limit of sin(x)/x as x approaches 0 is also equal to 1.
Therefore, lim (x→0) sin(x)/x = 1.
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