$\lim _{x \rightarrow 0}\left(\frac{\cot x}{\cot 2 x}\right)=2$
Answer (2)
Rewrite the cotangent functions in terms of sine and cosine: cot x = s i n x c o s x and cot 2 x = s i n 2 x c o s 2 x .
Apply the double angle identity sin 2 x = 2 sin x cos x and simplify the expression.
Evaluate the limit by direct substitution: lim x → 0 c o s 2 x 2 c o s 2 x .
The limit evaluates to 2 .
Explanation
Problem Setup We are asked to verify the limit x → 0 lim ( cot 2 x cot x ) = 2 To do this, we will rewrite the expression using trigonometric identities and then evaluate the limit.
Rewriting the Expression First, let's rewrite the cotangent functions in terms of sine and cosine: cot x = sin x cos x cot 2 x = sin 2 x cos 2 x Substituting these into the limit expression, we get: x → 0 lim ( s i n 2 x c o s 2 x s i n x c o s x ) = x → 0 lim ( sin x cos x ⋅ cos 2 x sin 2 x )
Applying Double Angle Identity Next, we use the double angle identity for sine, which states that sin 2 x = 2 sin x cos x . Substituting this into the expression, we have: x → 0 lim ( sin x cos x ⋅ cos 2 x 2 sin x cos x )
Simplifying the Expression Now, we can simplify the expression by canceling out the sin x terms: x → 0 lim ( cos 2 x 2 cos 2 x )
Evaluating the Limit Finally, we evaluate the limit by direct substitution. As x approaches 0, cos x approaches 1 and cos 2 x also approaches 1. Therefore, cos ( 2 ⋅ 0 ) 2 cos 2 ( 0 ) = 1 2 ( 1 ) 2 = 2
Final Answer Thus, the limit is indeed 2. x → 0 lim ( cot 2 x cot x ) = 2
Examples
Imagine you're designing a bridge and need to calculate the stability of a structure as it approaches a critical angle. The cotangent function is often used in these calculations, and understanding limits helps engineers determine the behavior of these functions near critical points, ensuring the bridge's safety and stability. This problem demonstrates a fundamental concept in calculus that is essential for solving real-world engineering problems.
To find lim x → 0 ( c o t 2 x c o t x ) , we rewrite cotangent in terms of sine and cosine, apply the double angle identity, simplify, and evaluate the limit. The result is 2 , verifying the original statement. Thus, the limit is 2 .
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