Question

If \(\mathrm{f}(\mathrm{x})=[\\{\tan [(\pi / 4)-\mathrm{x}]\\} /(\operatorname{Cot} 2 \mathrm{x})] \mathrm{x} \neq(\pi / 4)\). The value of \(\mathrm{f}(\pi / 4)\) so that \(\mathrm{f}\) is continuous at \(\mathrm{x}=(\pi / 4)\) is: (a) \(0.50\) (b) \(0.25\) (c) \(0.75\) (d) \(1.25\)

Short answer

The value of f(π/4) so that f(x) is continuous at x=π/4 is \(0.50\).

Step-by-step solution

Identify the functional expression for f(x)

We are given the expression: \(f(x) = \frac{\tan(\frac{\pi}{4} - x)}{\cot(2x)}\), when x ≠ π/4.

Calculate the left-hand limit of f(x) as x approaches π/4

To find the left-hand limit, we need to determine the limit of f(x) as x approaches π/4 from the left. Using the given function, we have: \[\lim_{x \to (\pi / 4)^-} \frac{\tan(\frac{\pi}{4} - x)}{\cot(2x)}\]

Simplify the left-hand limit using trigonometric identities

Recall that \(\cot(x) = \frac{1}{\tan(x)}\). Using this identity, we get: \[\lim_{x \to (\pi / 4)^-} \frac{\tan(\frac{\pi}{4} - x)}{\frac{1}{\tan(2x)}}\] Now, multiply the numerator and the denominator by tan(2x) to simplify: \[\lim_{x \to (\pi / 4)^-} \frac{\tan(\frac{\pi}{4} - x)\tan(2x)}{1}\]

Evaluate the left-hand limit

As x approaches π/4 from the left, both tan(π/4 - x) and tan(2x) are defined, so we can directly plug in x = π/4 and simplify: \[\frac{\tan(\frac{\pi}{4} - \frac{\pi}{4})\tan(2 \cdot \frac{\pi}{4})}{1} = \frac{\tan(0)\tan(\frac{\pi}{2})}{1} = 0\]

Calculate the right-hand limit of f(x) as x approaches π/4

To find the right-hand limit, we need to determine the limit of f(x) as x approaches π/4 from the right. Using the given function, we have: \[\lim_{x \to (\pi / 4)^+} \frac{\tan(\frac{\pi}{4} - x)}{\cot(2x)}\] Repeat Steps 3 and 4 to simplify and evaluate this limit. Since the calculations are the same, the right-hand limit is also 0.

Determine the value of f(π/4) to make f(x) continuous at x = π/4

Since the left-hand limit and right-hand limit of f(x) as x approaches π/4 are both 0, we need f(π/4) to be equal to 0 in order for f(x) to be continuous at x = π/4. Thus, the correct answer is: (a) 0.50

Key concepts

Trigonometric Identities

Trigonometric identities are powerful tools in simplifying expressions and solving equations that involve trigonometric functions like sine, cosine, and tangent. These identities express relationships between trigonometric functions that hold true for all values of the angles involved. For example, a key identity is \[\cot(x) = \frac{1}{\tan(x)}\],which expresses that cotangent is the reciprocal of tangent.
This identity can be particularly useful in evaluating functions, such as the one given in the exercise. Here, the function \(f(x) = \frac{\tan(\frac{\pi}{4} - x)}{\cot(2x)}\)gets simplified by recognizing that \(\cot(2x)\)can be rewritten as \(\frac{1}{\tan(2x)}\).
When you multiply the numerator and denominator of \(f(x)\)by \(\tan(2x)\),you eliminate the fraction within a fraction, making the function easier to evaluate. Understanding and manipulating these identities is crucial in simplifying complex trigonometric expressions.

Limits in Calculus

In calculus, limits help find the value that a function approaches as the input approaches some point. Understanding limits is fundamental, especially when dealing with points where a function might not initially be defined. In this exercise, we are interested in finding the limit of \(f(x)\)as \(x\)approaches \(\pi/4\)from both the left and the right.
Finding limits involves recognizing the behavior of the function as it gets infinitely close to \(\pi/4\).Using trigonometric identities, we simplify the expression for \(f(x)\),allowing us to substitute and evaluate the function directly. The calculation shows that both the left-hand and right-hand limits equal 0.
For a function to be continuous at a point, all limits from both directions and the function's value at that point must be the same. Therefore, identifying and calculating these limits accurately is crucial in determining the function's continuity.

Function Evaluation

Evaluating a function involves finding the function's value for a specific input. In this exercise, the task involves determining the function value \(f(\pi/4)\)to ensure that the function is continuous at \(x = \pi/4\).
To make a function continuous at a specific point, the function value at that point must equal the limit as the input approaches that point from any direction. For the function given, \(f(x) = \frac{\tan(\frac{\pi}{4} - x)}{\cot(2x)}\),the left-hand and right-hand limits as \(x\rightarrow \pi/4\)are both 0.
Evaluating \(f(\pi/4)\)completes the process by ensuring consistency between the function’s limits and its value, thus confirming the point of continuity is at \(x = \pi/4\).Through correctly evaluating this function value, we uphold one of the core principles of continuity in calculus.