The value of k which makes \(f\left( x \right)\; = \;\left\{ {\begin{array}{*{20}{c}} {\sin x\;,x \ne 0}\\ {k\;,x\; = \;0} \end{array}} \right.\) continuous at x = 0, is
0
Understanding Function Continuity at a Point
A function \(f(x)\) is said to be continuous at a point \(x = a\) if the following three conditions are met:
The function \(f(a)\) must be defined at \(x = a\).
The limit of the function as \(x\) approaches \(a\) must exist, i.e., \(\lim_{x \to a} f(x)\) must exist. This means the left-hand limit and the right-hand limit must be equal.
The limit of the function as \(x\) approaches \(a\) must be equal to the function value at \(a\), i.e., \(\lim_{x \to a} f(x) = f(a)\).
If any of these conditions are not satisfied, the function is discontinuous at \(x = a\).
Applying Continuity Conditions to the Given Function at x = 0
We are given the function: \(f\left( x \right)\; = \;\left\{ {\begin{array}{*{20}{c}} {\sin x\;,x \ne 0}\\ {k\;,x\; = \;0} \end{array}} \right.\) We need to find the value of \(k\) that makes this function continuous at \(x = 0\). Let's check the three conditions for continuity at \(x = 0\):
Condition 1: Function must be defined at x = 0
According to the definition of the function, when \(x = 0\), \(f(x) = k\). So, \(f(0) = k\). For \(f(0)\) to be defined, \(k\) must be a finite real number. This condition is met as long as \(k\) is a specific value.
Condition 2: The limit of the function as x approaches 0 must exist
We need to find \(\lim_{x \to 0} f(x)\). As \(x\) approaches 0, \(x\) is not equal to 0. Therefore, we use the definition of \(f(x)\) for \(x \ne 0\), which is \(\sin x\). So, we need to evaluate \(\lim_{x \to 0} \sin x\). The sine function is continuous everywhere, so the limit as \(x\) approaches 0 is simply the value of the function at \(x = 0\). \(\lim_{x \to 0} \sin x = \sin 0\) We know that \(\sin 0 = 0\). Therefore, \(\lim_{x \to 0} f(x) = 0\). The limit exists and is equal to 0.
Condition 3: The limit must equal the function value at x = 0
For the function to be continuous at \(x = 0\), the limit of the function as \(x\) approaches 0 must be equal to the function value at \(x = 0\). Mathematically, \(\lim_{x \to 0} f(x) = f(0)\). From Condition 2, we found \(\lim_{x \to 0} f(x) = 0\). From Condition 1, we know \(f(0) = k\). Equating these two values: \(0 = k\)
Determining the Value of k for Continuity
For the function \(f(x)\) to be continuous at \(x = 0\), the value of \(k\) must be 0. This makes the limit of the function at \(x=0\) equal to the function's value at \(x=0\), satisfying all conditions for continuity.
Let's summarize the findings:
| Condition | Requirement at \(x = 0\) | Result | Met? |
|---|---|---|---|
| 1. \(f(0)\) defined | \(f(0) = k\) must be a finite value | \(f(0) = k\) | Yes (if \(k\) is finite) |
| 2. \(\lim_{x \to 0} f(x)\) exists | \(\lim_{x \to 0, x \ne 0} \sin x\) must exist | \(\lim_{x \to 0} \sin x = \sin 0 = 0\) | Yes (Limit is 0) |
| 3. \(\lim_{x \to 0} f(x) = f(0)\) | \(0 = k\) | \(k = 0\) | Yes (if \(k=0\)) |
Therefore, the value of \(k\) that ensures the function is continuous at \(x=0\) is 0.
Revision Table: Key Concepts for Function Continuity
| Concept | Definition/Explanation |
|---|---|
| Continuity at a Point | A function \(f(x)\) is continuous at \(x=a\) if \(\lim_{x \to a} f(x) = f(a)\). This implies \(f(a)\) is defined and the limit exists. |
| Limit of a Function | The value that \(f(x)\) approaches as \(x\) gets arbitrarily close to \(a\), but not necessarily equal to \(a\). Notation: \(\lim_{x \to a} f(x)\). |
| Left-Hand Limit | The value \(f(x)\) approaches as \(x\) approaches \(a\) from values less than \(a\) (\(x < a\)). Notation: \(\lim_{x \to a^-} f(x)\). |
| Right-Hand Limit | The value \(f(x)\) approaches as \(x\) approaches \(a\) from values greater than \(a\) (\(x > a\)). Notation: \(\lim_{x \to a^+} f(x)\). |
| Existence of Limit | \(\lim_{x \to a} f(x)\) exists if and only if \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)\). |
| Piecewise Function | A function defined by multiple sub-functions, each applying to a different interval of the domain. Continuity at the points where the definition changes requires special checking using limits. |
Additional Information on Function Continuity and Limits
Continuity is a fundamental concept in calculus. A function that is continuous on an interval can be drawn without lifting the pen. Discontinuities can occur for various reasons, such as holes, jumps, or vertical asymptotes.
For piecewise functions, like the one in this question, checking continuity at the points where the definition changes is crucial. In this case, the definition changes at \(x=0\), which is why we specifically checked continuity there.
The function \(f(x) = \sin x\) is known to be continuous for all real numbers. This property allowed us to easily evaluate the limit \(\lim_{x \to 0} \sin x\) by direct substitution.
If the limit \(\lim_{x \to 0} f(x)\) had resulted in a value different from 0, or if the limit did not exist, then the function could not be made continuous at \(x=0\) regardless of the value of \(k\). The fact that the limit exists and is a finite value (0 in this case) allows us to find a specific value of \(k\) (also 0) that 'fills the gap' or matches the limit, thereby ensuring continuity.
Similar Questions
A function is defined as follows: \[ f(x) = \begin{cases} -\dfrac{x}{\sqrt{x^2}}, & x \ne 0, \\[6pt] 0, & x = 0. \end{cases} \] Which one of the following is correct in respect of the above function?
Let f : A → R, where A = R\(0) is such that \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{\rm{x}} + \left| {\rm{x}} \right|}}{{\rm{x}}}\) . On which one of the following sets is f(x) continuous?
Consider the following statements for f(x) = e -|x| ;
1. The function is continuous at x = 0.
2. The function is differentiable at x = 0.
Which of the above statements is / are correct?
If the function \(\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a + bx,\;\;}&{x < 1}\\ {5,}&{x = 1}\\ {b - ax,}&{x > 1} \end{array}} \right.\) is continuous, then what is the value of (a + b)?
The function \({\rm{f}}\left( {\rm{x}} \right) = \frac{{1 - \sin {\rm{x}} + \cos {\rm{x}}}}{{1 + \sin {\rm{x}} + \cos {\rm{x}}}}\) is not defined at x = π. The value of f(π) so that f(x) is continuous at x = π, is
Consider the following statements in respect of the function \(\rm f(x) = sin \left(\frac{1}{x^2}\right)\) , x ≠ 0:
1. It is continuous at x = 0, if f(0) = 0.
2. It is continuous at \(x = \frac{2}{\sqrt{\pi}}\) .
Which of the above statements is/are correct?
If \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{{\rm{x}}^2} - 9}}{{{{\rm{x}}^2} - 2{\rm{x}} - 3}}\) , x ≠ 3 is continuous at x = 3, then which one of the following is correct?
Let f(x) be defined as follows: \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {2{\rm{x}} + 1,{\rm{\;\;}} - 3 < {\rm{x}} < - 2}\\ {{\rm{x}} - 1,{\rm{\;\;}} - 2 \le {\rm{x}} < 0}\\ {{\rm{x}} + 2,{\rm{\;\;\;}}0 \le {\rm{x}} < 1} \end{array}} \right.\) Which one of the following statements is correct in respect of the above function?
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{{\rm{e}}^{\rm{x}}} - 1}}{{\rm{x}}},{\rm{\;\;x}} > 0}\\ {0,{\rm{\;\;\;x}} = 0} \end{array}} \right.\)
Which one of the following statements is correct?
\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{{\rm{e}}^{\rm{x}}} - 1}}{{\rm{x}}},{\rm{\;\;x}} > 0}\\ {0,{\rm{\;\;\;x}} = 0} \end{array}} \right.\)
Which of the following statements is/are correct?
1. f(x) is right continuous at x = 0
2. f(x) is discontinuous at x = 1.
Select the correct answer using the code given below:
Important Questions from Continuity of a function
A function is defined as follows: \[ f(x) = \begin{cases} -\dfrac{x}{\sqrt{x^2}}, & x \ne 0, \\[6pt] 0, & x = 0. \end{cases} \] Which one of the following is correct in respect of the above function?
f(x) = x + |x| is continuous for
Let f : A → R, where A = R\(0) is such that \({\rm{f}}\left( {\rm{x}} \right) = \frac{{{\rm{x}} + \left| {\rm{x}} \right|}}{{\rm{x}}}\) . On which one of the following sets is f(x) continuous?
Consider the following statements for f(x) = e -|x| ;
1. The function is continuous at x = 0.
2. The function is differentiable at x = 0.
Which of the above statements is / are correct?
If the function \(\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a + bx,\;\;}&{x < 1}\\ {5,}&{x = 1}\\ {b - ax,}&{x > 1} \end{array}} \right.\) is continuous, then what is the value of (a + b)?
