Question

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow \pi / 4} 3 \csc 2 x \cot 2 x$$

Short answer

Question: Evaluate the limit of the expression as x approaches π/4: $$\lim_{x\rightarrow \pi/4} 3\csc 2x \cot 2x$$ Answer: The limit of the given expression as x approaches π/4 is 6.

Step-by-step solution

Rewrite the given expression using trigonometric identities.

We can rewrite the given expression using the reciprocal and double angle identities. The reciprocal identity for cosecant and cotangent are \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\), and the double angle identity is \(\sin 2x = 2\sin x \cos x\) and \(\cos 2x = \cos^2 x - \sin^2 x\). Applying these identities, we get: $$3\csc 2x \cot 2x = 3\cdot \frac{1}{\sin 2x} \cdot \frac{\cos 2x}{\sin 2x}$$

Simplify the expression and cancel out common factors.

Simplifying the expression from Step 1, we get: $$3\cdot \frac{1}{\sin 2x} \cdot \frac{\cos 2x}{\sin 2x} = \frac{3\cos 2x}{(\sin 2x)^2}$$

Rewrite the expression using double angle identities.

Now, rewrite the expression using double angle identities for both sine and cosine: $$\frac{3\cos 2x}{(\sin 2x)^2} = \frac{3(\cos^2 x - \sin^2 x)}{(2\sin x \cos x)^2}$$

Simplify the expression using the Pythagorean identity.

We know that the Pythagorean identity states that \(\sin^2 x + \cos^2 x = 1\). Therefore, we can rewrite \(\cos^2 x\) as \(1 - \sin^2 x\). Substituting this into the expression gives: $$\frac{3(1 - \sin^2 x - \sin^2 x)}{(2\sin x \cos x)^2} = \frac{3(1 - 2 \sin^2 x)}{(2\sin x \cos x)^2}$$

Evaluate the limit as x approaches π/4.

Now, we can evaluate the limit as \(x \rightarrow \pi/4\). Substituting \(\pi/4\) into the expression, we get: $$\lim _{x \rightarrow \pi / 4} \frac{3(1 - 2 \sin^2 x)}{(2\sin x \cos x)^2} = \frac{3(1 - 2 \sin^2 (\pi/4))}{(2\sin (\pi/4) \cos (\pi/4))^2}$$ Evaluate the sine and cosine at \(\pi/4\), which gives us \(\sin (\pi/4) = \cos (\pi/4) = \frac{1}{\sqrt{2}}\). Substituting these values, we get: $$\frac{3(1 - 2 (\frac{1}{2})^2)}{(2\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}})^2}$$

Simplify and calculate the limit.

Simplifying the expression, we get: $$\frac{3(1 - 2 (\frac{1}{4}))}{(2(\frac{1}{2}))^2} = \frac{3(1 - \frac{1}{2})}{(\frac{1}{2})^2} = \frac{3\left(\frac{1}{2}\right)}{\left(\frac{1}{4}\right)} = 3\left(\frac{1}{2}\right)(4) = 3(2) = 6$$ So, the limit as x approaches \(\pi/4\) of the given expression is: $$\lim_{x\rightarrow \pi/4} 3\csc 2x \cot 2x = \boxed{6}$$

Key concepts

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that hold true for all values of the variables where both sides of the equality are defined. These identities are invaluable tools for simplifying and evaluating expressions involving trigonometric functions. Among the most commonly used sets of identities are the reciprocal identities, such as \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \).

Understanding how to apply these identities to reformulate expressions is a key skill in calculus, particularly when evaluating limits. For instance, in the example provided, the reciprocal identities were used to express \( \csc 2x \) and \( \cot 2x \) in terms of sine and cosine, allowing for further manipulation and simplification of the expression. This step is crucial, as it paves the way for applying other trigonometric identities and eventually reaching a form where the limit can be directly evaluated.

Double Angle Identities

Double angle identities express trigonometric functions of twice an angle in terms of functions of the original angle. For the functions sine and cosine, the respective double angle identities are \( \sin 2x = 2\sin x \cos x \) and \( \cos 2x = \cos^2 x - \sin^2 x \).

These identities are particularly useful when an expression involves trigonometric functions of twice an angle, as seen in the step-by-step solution provided for \( \lim _{x \rightarrow \pi / 4} 3 \csc 2x \cot 2x \). By employing double angle identities, the expression in the limit problem was simplified into a form that revealed a quotient involving squares of sine and cosine functions. This opens up an opportunity to further simplify the expression using the Pythagorean identity, showing how overlapping identities can collaboratively lead to a solution.

Pythagorean Identity

The Pythagorean identity, one of the fundamental identities in trigonometry, is derived from the Pythagorean theorem. It states that for any angle \( x \), \( \sin^2 x + \cos^2 x = 1 \). This identity illustrates the intrinsic relationship between the sine and cosine of an angle.

In the context of the example, this relationship allowed us to replace \( \cos^2 x \) with \( 1 - \sin^2 x \) to simplify the expression. By doing so, we brought the expression one step closer to a form that made evaluation of the limit straightforward when \( x \rightarrow \pi/4 \). The Pythagorean identity frequently serves as the last piece of the puzzle in some limit problems, as it helps to eliminate complex trigonometric terms in favor of simpler algebraic ones, leading to a limit value that can be easily computed.