Evaluate the limit.
The limit as x approaches 0 of
(x-sin(x))/(x-tan(x))
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To evaluate this limit, we can use algebraic manipulation and trigonometric identities. Let's break it down step by step:
Step 1: Factor out the common factor of x in the numerator and denominator:
(x - sin(x))/(x - tan(x)) = (x(1 - sin(x)/x))/(x(1 - tan(x)/x))
Step 2: Simplify the expression by canceling out the x terms in the numerator and denominator:
(x - sin(x))/(x - tan(x)) = (1 - sin(x)/x)/(1 - tan(x)/x)
Step 3: Apply the trigonometric identity sin(x)/x ≈ 1 for small values of x.
Since we are taking the limit as x approaches 0, we can approximate sin(x)/x as 1.
Using this approximation, the expression becomes:
(1 - 1)/(1 - tan(x)/x)
Step 4: Simplify the expression further:
(1 - 1)/(1 - tan(x)/x) = 0/(1 - tan(x)/x) = 0/1 = 0
Therefore, the limit as x approaches 0 of (x - sin(x))/(x - tan(x)) is 0.