91Ó°ÊÓ

Match the entries of the following two columns.Column I \(\quad\) Column \(=\) II \hline (A) \(f(x)=\left[\begin{array}{ll}x+1, & \text { if } x<0 \\ \cos x, & \text { if } x \geq 0\end{array}\right.\) at \(x\) t \(x=0\) is (p) continuous (B) For every \(x \in R\), the function \(g(x)=\frac{\sin (\pi[x-\pi])}{1+[x]^{2}}\) (q) differentiability where \([x]\) denotes the greatest integer function is (C) \(h(x)=\sqrt{\\{x\\}^{2}}\) where \(\\{x\\}\) denotes fractional part ( \(\left.\mathrm{r}\right)\) discontinuous (r) function for all \(x \in I\), is \(k(x)=\left\\{\begin{array}{cl}x^{\frac{1}{\ln x}}, & \text { if } x \neq 1 \\\ e, & \text { if } x=1\end{array}\right.\) (D) \(k(x)=\left\\{x^{\ln x}\right.\), if \(x \neq 1\) at \(x=1\) is at \(x=1\) (s) non-derivable

Step by step solution

01

Identify Function Behavior at x=0

For the piecewise function \( f(x) \), consider each condition separately. When \( x < 0 \), \( f(x) = x + 1 \), and when \( x \geq 0 \), \( f(x) = \cos(x) \). Evaluate continuity at \( x = 0 \) by checking if the left-hand limit, right-hand limit, and the function value at zero are equal.

02

Evaluate Continuity of f(x)

Find \( \lim_{x \to 0^-} f(x) = 0 + 1 = 1 \) and \( \lim_{x \to 0^+} \cos(x) = \cos(0) = 1 \). Since \( f(0) = \cos(0) = 1 \), the function is continuous at \( x = 0 \). Therefore, A matches with (p) **continuous**.

03

Determine Behavior of g(x)

The function \( g(x) = \frac{\sin(\pi[x-\pi])}{1+[x]^2} \) involves the greatest integer function, \( [x] \). For any real \( x \), \([x] \) is constant over intervals, making the numerator and denominator piecewise constant. Evaluate the continuity and differentiability, realizing \( [x] \) creates discontinuities.

04

Evaluate Discontinuity of g(x)

Since \( [x] \) is discontinuous at integer values, \( g(x) \) is also discontinuous at these points. As none of the conditions for continuity hold everywhere, B matches with (r) **discontinuous**.

05

Analyze Square Root of Fractional Part Function

For \( h(x) = \sqrt{\{x\}^2} \), where \( \{x\} \) is the fractional part, \( h(x) = \{x\} \) behaves continuously over any interval \([n, n+1)\) since the fractional part function is continuous within these intervals.

06

Evaluating Continuous Behavior of h(x)

Since the fractional part is continuous and \( h(x) = \sqrt{\{x\}^2} = \{x\} \), it remains continuous over real numbers. Therefore, (\{x\} is known to be continuous over non-integer values but discontinuous between segments), this doesn't match any discontinuity, hence ignore further attention.

07

Examine Continuity of k(x) at x=1

For \( k(x) = x^{\frac{1}{\ln x}} \) for \( x eq 1 \) and \( k(1) = e \), check the limit as \( x \to 1 \). Direct calculation at \( x=1 \) involves \( \ln(x) = \ln(1) = 0 \), hence problematic for defining fractional power without Taylor or special knowledge approximation.

08

Asses Continuity at x=1 for k(x)

Direct substitution and behavior at \( k(1) = e \) is matched if considering indeterminacy and considering limits not directly addressing, focusing instead on "is this differentiable?" identifying relating limit confirms differentiability error not directly for easy solution steps.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Piecewise Functions

Piecewise functions are a special type of function that has different expressions based on the input value. They are defined in segments or pieces. For example, the function \( f(x) \) in our exercise is defined by two different expressions: \( x + 1 \) when \( x < 0 \) and \( \cos(x) \) when \( x \geq 0 \). Piecewise functions can have breaks or different behaviors at the boundaries, creating unique challenges when analyzing them for continuity or differentiability.
When dealing with piecewise functions, it's important to:

• Check the behavior of each piece individually.
• Evaluate the function at the boundary values.
• Assess both limits to determine continuity or differentiability at the break points.

Understanding piecewise functions helps us study more complex mathematical problems that can't be represented by a single formula.

Greatest Integer Function

The greatest integer function, often denoted by \( [x] \), maps any real number to the largest integer less than or equal to that number. This function introduces steps, much like a staircase, contributing to discontinuity where \( x \) is an integer.
For example, \( g(x) = \frac{\sin(\pi[x-\pi])}{1+[x]^2} \) uses the greatest integer function to evaluate discontinuity since \( [x] \) has jumps at integer values.
Key characteristics of the greatest integer function include:

• Outputs integers for any real input.
• Creates discontinuities at integer points.
• Segments behavior between consecutive integers.

This function is immensely useful in areas of algorithms and computer science where discrete methods are applied.

Fractional Part Function

The fractional part function, \( \{x\} \), gives the non-integer part of a real number. Defined as \( \{x\} = x - [x] \), it represents the remaining part of \( x \) after removing the largest integer less than or equal to \( x \).
In the context of the provided exercise, \( h(x) = \sqrt{\{x\}^2} = \{x\} \) utilizes the fractional part function. It maintains continuity within any interval \( [n, n+1) \) since the fractional part smoothly varies between \( 0 \) and \( 1 \).
Properties of the fractional part function:

• It is periodic with period 1.
• A continuous function for non-integers.
• Discontinuous at integer points where it drops back to 0.

Understanding the fractional part is crucial for problems involving precision or modulo operations.

Discontinuity

Discontinuity occurs in a function when there is an interruption or jump in its graph. There are several types of discontinuities, but the most common one associated with piecewise, greatest integer, and fractional part functions is the jump discontinuity.
For instance, the function \( g(x) = \frac{\sin(\pi[x-\pi])}{1+[x]^2} \) experiences discontinuities at integer values of \( x \). This occurs as the greatest integer function shifts from one integer value to the next.
Keep in mind:

• Discontinuities affect limits, making differentiation and integration challenging at these points.
• Analyzing left-hand and right-hand limits helps identify the discontinuity type.
• Continuity can be restored across points by re-defining function sections.

Recognizing and dealing with discontinuities is essential for studying advanced calculus and real analysis.

Limits and Continuity

Limits and continuity are foundational concepts in calculus that describe the behavior of functions as inputs approach a particular value. A function is continuous at a point if the limit of the function as it approaches from both sides equals the function's value at that point.
In our exercise, evaluating the continuity of \( f(x) \) involves checking \( \lim_{x \to 0^-} f(x) \), \( \lim_{x \to 0^+} f(x) \), and \( f(0) \). Since these values are the same, \( f(x) \) is continuous at \( x=0 \).
Vital concepts to grasp include:

• Limits are used to evaluate the behavior of functions at edges or critical points.
• Continuity implies no breaks, jumps, or holes at a point.
• Both left and right-hand limits should coincide with the function's value for continuity.

Understanding these principles is critical for problem-solving in calculus.

Differentiability

Differentiability is the property of a function that indicates the presence of a derivative at each point in its domain. If a function is differentiable at a point, it means the graph has a defined tangent there without any sharp turns or cusps.
In the problem, studying \( k(x) = x^{\frac{1}{\ln x}} \) at \( x=1 \) reveals challenges in finding limits, indicating potential issues with differentiability. Differentiability signifies a function’s smoothness, and it inherently implies continuity at that point.
Key considerations include:

• Differentiability implies continuity.
• Functions are non-differentiable at points of discontinuity or sharp edges.
• Explicit conditions such as \( \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} \) define differentiability.

Gaining insights into differentiability aids in the assessment of function behavior and in solving higher-order calculus problems.

Indeterminate Forms

Indeterminate forms arise in calculus when limits result in expressions that are not directly solvable, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms occur frequently in problems involving limits or during differentiation.
At \( x=1 \), \( k(x) \) exhibited a form involving \( \ln(x) \) causing indeterminacy as \( x ightarrow 1 \). Indeterminate forms require special techniques like L'Hôpital's rule or Taylor series to find the limit.
Considerations include:

• Identifying when expressions result in indeterminate forms.
• Employing calculus techniques to resolve them into solvable forms.
• Understanding the nature of functions causing these forms to appear.

Mastery of indeterminate forms is indispensable for tackling complex limit problems confidently.