Evaluate the limit as x approaches 0 of the expression [tex]\( \frac{\cos x \sin^2 x - 1 + \cos^2 x}{2x^4} \)[/tex].
Answer (2)
The limit as x approaches 0 of the expression 2 x 4 c o s x s i n 2 x − 1 + c o s 2 x is 6 − 1 . This was found by using Taylor series expansions to simplify the expression, allowing us to evaluate the limit as x approaches 0. The steps involved resolving an indeterminate form and applying series approximations.
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To evaluate the limit as x approaches 0 of the expression 2 x 4 cos x sin 2 x − 1 + cos 2 x , let's go through the calculation step-by-step.
Step 1: Simplify the Expression
The given expression is: cos x sin 2 x − 1 + cos 2 x .
Notice that cos 2 x + sin 2 x = 1 . So, cos 2 x = 1 − sin 2 x .
Substitute cos 2 x in the expression: cos x sin 2 x − 1 + ( 1 − sin 2 x ) = cos x sin 2 x − 1 + 1 − sin 2 x .
Simplify: cos x sin 2 x − sin 2 x .
Factor out sin 2 x :
sin 2 x ( cos x − 1 ) .
Step 2: Consider the Limit
Now, rewrite the entire expression: 2 x 4 sin 2 x ( cos x − 1 ) .
As x → 0 , sin x ≈ x and cos x − 1 ≈ − 2 x 2 using the small-angle approximations.
Therefore, near x = 0 :
sin 2 x ≈ x 2 and ( cos x − 1 ) ≈ − 2 x 2 .
Substituting these into the expression: 2 x 4 x 2 ⋅ ( − 2 x 2 ) = 2 x 4 − 2 x 4 .
Step 3: Simplify Further
Simplifying: 4 x 4 − x 4 = − 4 1 .
Thus, the limit is − 4 1 .
This result shows the behavior of the original expression as x → 0 by using both trigonometric identities and small-angle approximations.
