Q.If a function defined by 𝑓(𝑥) = { 𝑘𝑥 + 1, 𝑥 ≤ 𝜋 cos 𝑥 , 𝑥 > 𝜋 is continuous at 𝑥 = 𝜋, then the value of 𝑘 is
(A) 𝜋
(B) −1 𝜋
(C) 0
(D) −2 𝜋

Concept and intuition

The equations $x = t - \sin t$, $y = 1 - \cos t$ describe a cycloid. Derivatives with respect to $x$ are computed via parametric formulas:

The condition $\frac{d^2y}{dx^2} = -1$ gives an equation for $t$, which we solve for $K>0$. Then $\lim_{t\to K} y/x$ is simply $y(K)/x(K)$ because both are continuous at $K$ (and $x(K)\neq 0$).

Step-by-step solution

1. Compute first derivatives

Hence

2. Simplify $\frac{dy}{dx}$ using a half-angle identity Recall $1 - \cos t = 2\sin^2(t/2)$ and $\sin t = 2\sin(t/2)\cos(t/2)$. Then

provided $\sin(t/2) \neq 0$ (which holds for $t>0$ not a multiple of $2\pi$).

3. Compute $\frac{d^2y}{dx^2}$ First,

Then

Using $1 - \cos t = 2\sin^2(t/2)$, we get

4. Apply the given condition Set $\frac{d^2y}{dx^2} = -1$:

Hence $\csc^2(t/2) = 2$ (positive), so $\sin^2(t/2) = 1/2$, i.e. $\sin(t/2) = 1/\sqrt{2}$ (since $t>0$ and we take the positive root for the smallest positive $K$). …

The key idea is that a piecewise function is continuous everywhere if and only if it is continuous at each boundary where the definition changes. At those points, the left-hand limit, right-hand limit, and the function’s value must all be equal. Here we have three boundaries: $x=0$, $x=1$, and $x=2$. Each gives one equation, and three unknowns ($a,b,c$) means we can solve uniquely.

Let’s work through each boundary carefully.

1. Continuity at $x=0$ For $x \leq 0$, $f(x) = \cos(2x + \pi)$. At $x=0$, this gives $f(0) = \cos(\pi) = -1$. For $0 < x < 1$, $f(x) = ax^2 + b$. As $x \to 0^+$, this approaches $a(0)^2 + b = b$. Continuity at $x=0$ requires

so $-1 = b$. Hence

2. Continuity at $x=1$ For $0 < x < 1$, $f(x) = ax^2 + b$. As $x \to 1^-$, this approaches $a(1)^2 + b = a + b$. For $1 \leq x \leq 2$, $f(x) = cx + 4$. At $x=1$, this gives $f(1) = c(1) + 4 = c + 4$. Continuity at $x=1$ requires

so $a + b = c + 4$. Substituting $b = -1$ gives

3. Continuity at $x=2$ For $1 \leq x \leq 2$, $f(x) = cx + 4$. As $x \to 2^-$, this approaches $c(2) + 4 = 2c + 4$. For $x \geq 2$, $f(x) = 3a + 1$. At $x=2$, this gives $f(2) = 3a + 1$. Continuity at $x=2$ requires

so $2c + 4 = 3a + 1$. Rearranging:

Now solve the system:

A function is continuous everywhere if it is continuous at every point in its domain. For a piecewise function, the individual pieces are typically continuous within their defined intervals (polynomials and trigonometric functions are continuous). The critical points to check for continuity are the boundary points where the function's definition changes.

In this problem, the function $f(x)$ is defined differently for $x \leq 0$, $0 < x \leq 2$, and $x > 2$. The potential points of discontinuity are $x=0$ and $x=2$. For $f(x)$ to be continuous everywhere, it must be continuous at these two points.

We will apply this condition at $x=0$ and $x=2$ to find the values of $a$ and $b$.

1. Check continuity at $x=0$: For $f(x)$ to be continuous at $x=0$, the left-hand limit, right-hand limit, and the function value at $x=0$ must all be equal.
   • Left-hand limit at $x=0$: For $x < 0$, $f(x) = 1 + \cos x$.

*   **Right-hand limit at $x=0$**: For $x > 0$, $f(x) = a - x$.

*   **Function value at $x=0$**: For $x \leq 0$, $f(x) = 1 + \cos x$.

For continuity at $x=0$, we must have $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$.
Therefore, $2 = a = 2$. This implies that $a=2$.

2. Check continuity at $x=2$:

Similarly, for $f(x)$ to be continuous at $x=2$, the left-hand limit, right-hand limit, and the function value at $x=2$ must all be equal.

* Left-hand limit at $x=2$: For $x < 2$, $f(x) = a - x$. …

Concept & Intuition

A piecewise function is continuous on its domain if it doesn’t “jump” at the points where the formula changes. At each such boundary, the left-hand limit (from the piece before) must equal the right-hand limit (from the piece after), and both must equal the function’s value there (if defined). Here the boundaries are $x=1$ and $x=4$. We compute the limits using standard calculus facts: $\lim_{u\to0} \frac{\tan u}{u}=1$ and $\lim_{u\to0} \frac{b^u-1}{u} = \log b$ (but careful — the third piece is $\frac{b^x-1}{x}$ as $x\to4^+$, not near 0, so we just plug in directly). The second piece is a rational function that simplifies.

Step-by-step solution

1. Continuity at $x=1$ For $0<x<1$, $f(x) = \frac{\tan a(x-1)}{x-1}$. Let $u = x-1$; as $x\to1^-$, $u\to0^-$. Then

For $1\le x\le4$, at $x=1$ we have

Continuity requires $\lim_{x\to1^-} f(x) = f(1)$, so

$a = \frac{31}{6}.$ …

We evaluate $\displaystyle \lim_{x\to 3} \frac{11 - [2-x]}{[x+10]}$, where $[t]$ is the greatest integer $\le t$.

Left-hand limit ($x \to 3^-$): take $x$ slightly less than $3$ (e.g. $x = 2.9$).

So the expression is $\dfrac{11 - (-1)}{12} = \dfrac{12}{12} = 1.$

Right-hand limit ($x \to 3^+$): take $x$ slightly more than $3$ (e.g. $x = 3.1$). …

Concept and intuition:

The greatest integer function $[t]$ (also called the floor function) jumps at every integer. When we take a limit as $x$ approaches a point, if the argument of the floor function passes through an integer, the floor value changes abruptly. That often makes the two-sided limit fail to exist. Here, as $x \to 3$, both $[2 - x]$ and $[x + 10]$ cross integer boundaries, so we must check the left and right separately.

1. Identify the critical integer boundaries.

   For $x \to 3$, consider the expressions inside the floor functions:

   • $2 - x$: when $x = 3$, $2 - 3 = -1$. For $x$ just less than 3, $2 - x > -1$; for $x$ just greater than 3, $2 - x < -1$. So the floor $[2 - x]$ will change at $x = 3$.
   • $x + 10$: when $x = 3$, $x + 10 = 13$. For $x$ near 3, $x+10$ is near 13, but does it cross 13? Since 13 is an integer, we need to see if $x+10$ passes through 13 from below or above as $x$ passes through 3. Actually, $x+10 = 13$ exactly at $x=3$. For $x < 3$, $x+10 < 13$; for $x > 3$, $x+10 > 13$. So $[x+10]$ also changes at $x=3$.

2. Compute the left-hand limit ($x \to 3^-$).

   Let $x = 3 - h$ with $h \to 0^+$.

   • $2 - x = 2 - (3 - h) = -1 + h$. Since $h > 0$ small, $-1 + h$ is slightly greater than $-1$ but less than 0. Hence $[2 - x] = -1$.
   • $x + 10 = (3 - h) + 10 = 13 - h$. Since $h > 0$, $13 - h$ is slightly less than 13, so $[x + 10] = 12$.
   • The expression becomes $\frac{11 - (-1)}{12} = \frac{12}{12} = 1$.

3. Compute the right-hand limit ($x \to 3^+$).

   Let $x = 3 + h$ with $h \to 0^+$.

   • $2 - x = 2 - (3 + h) = -1 - h$. This is slightly less than $-1$, so $[2 - x] = -2$.
   • $x + 10 = (3 + h) + 10 = 13 + h$. This is slightly greater than 13, so $[x + 10] = 13$.
   • The expression becomes $\frac{11 - (-2)}{13} = \frac{13}{13} = 1$. …

When evaluating limits, especially those involving differentiable functions, the first step is always to substitute the limiting value to check for indeterminate forms. If we encounter $\frac{0}{0}$ or $\frac{\infty}{\infty}$, it indicates that further analysis is required, often involving L'Hôpital's Rule or algebraic manipulation to use the definition of the derivative.

The definition of the derivative is a fundamental concept:

Our strategy will be to transform the given limit expression into a form that allows us to directly apply this definition. This usually involves adding and subtracting a specific term in the numerator to create the necessary $(f(x) - f(a))$ and $(x-a)$ components.

Here's how we can solve the problem:

1. Check for indeterminate form:

   Let's substitute $x=a$ into the given expression:

   Numerator: $af(a) - af(a) = 0$

   Denominator: $a-a = 0$

   Since we have the indeterminate form $\frac{0}{0}$, we can proceed with further evaluation.

2. Manipulate the numerator:

   The numerator is $xf(a) - af(x)$. To use the definition of the derivative, we need terms like $f(x) - f(a)$. Notice that we have $af(x)$. If we could pair it with $af(a)$, we would get $a(f(x) - f(a))$.

   Let's add and subtract $af(a)$ in the numerator:

Now, we can factor terms:

3. Substitute back into the limit and split: Substitute this manipulated numerator back into the limit expression:

Now, split the fraction into two separate terms:

As $x \to a$, $x-a \neq 0$, so we can cancel the $(x-a)$ term in the first part:

4. Evaluate the limits: Using the properties of limits, we can evaluate each part separately: …

We need to find

Direct substitution gives $\frac{0}{0}$, so we must use series expansions. The key idea: expand each term as a power series in $x$, keep terms up to $x^2$, and simplify.

1. Expand $2^{2x}$ Recall $a^{bx} = e^{bx \log a}$. So

That is

2. Expand $2^{x+1}$ Write $2^{x+1} = 2 \cdot 2^x = 2 e^{x \log 2}$. Then

3. Expand $\cos 2x$

4. Assemble the numerator The numerator is

Substitute expansions:

Combine constants: $1 - 2 + 2 - 1 = 0$.

Combine $x$ terms: $2(\log 2)x - 2(\log 2)x = 0$.

Combine $x^2$ terms: $2(\log 2)^2 x^2 - (\log 2)^2 x^2 + 2x^2 = (\log 2)^2 x^2 + 2x^2$.

So

$N = \bigl[2 + (\log 2)^2\bigr] x^2 + O(x^3).$ …

Concept & Intuition

When $x \to 0$, both $\sin x$ and $\tan x$ go to $0$, so $3^{\sin x} \to 3^0 = 1$ and $2^{\tan x} \to 1$. The numerator becomes $1 - 1 = 0$, denominator also $0$, so we have a $\frac{0}{0}$ form.

The trick: write $a^{f(x)} = e^{f(x) \log a}$. Then near $0$, $e^{u} - 1 \approx u$, which lets us replace exponentials by linear approximations. This reduces the limit to a combination of $\frac{\sin x}{x}$ and $\frac{\tan x}{x}$ limits.

Step-by-step solution

1. Rewrite the exponentials

So the numerator becomes

2. Add and subtract 1 to create a standard form Write

This is valid because $1-1=0$.

3. Split into two limits