Evaluate the limit Lim (tan(x)/sin(x))x→0
To evaluate the limit as x approaches 0 of (tan(x) / sin(x)), we can use L’Hopital’s Rule or rewrite the expression in terms of sine and cosine
To evaluate the limit as x approaches 0 of (tan(x) / sin(x)), we can use L’Hopital’s Rule or rewrite the expression in terms of sine and cosine.
Method 1: L’Hopital’s Rule
Applying L’Hopital’s Rule entails taking the derivative of the numerator and the derivative of the denominator separately until we no longer have an indeterminate form (0/0).
Take the derivatives:
lim(x→0) [(d/dx) tan(x)] / [(d/dx) sin(x)]
The derivative of tan(x) is sec^2(x), and the derivative of sin(x) is cos(x).
lim(x→0) [sec^2(x)] / [cos(x)]
Now, substitute x = 0 into the expression:
[sec^2(0)] / [cos(0)]
sec(0) = 1 and cos(0) = 1, so the limit simplifies to:
1 / 1 = 1
Therefore, the limit of (tan(x) / sin(x)) as x approaches 0 is equal to 1.
Method 2: Rewrite in terms of sine and cosine
We can also rewrite (tan(x) / sin(x)) using identities involving sine and cosine.
tan(x) = sin(x) / cos(x)
Substituting this into the expression:
lim(x→0) [(sin(x) / cos(x)) / sin(x)]
The sin(x) terms in the numerator cancel out:
lim(x→0) [1 / cos(x)]
Now, substitute x = 0 into the expression:
1 / cos(0)
Using the identity cos(0) = 1, we get:
1 / 1 = 1
Therefore, using either method, the limit of (tan(x) / sin(x)) as x approaches 0 is equal to 1.
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