Advertisements
Advertisements
Question
The value of a for which the function \[f\left( x \right) = \begin{cases}\frac{\left( 4^x - 1 \right)^3}{\sin\left( x/a \right) \log \left\{ \left( 1 + x^2 /3 \right) \right\}}, & x \neq 0 \\ 12 \left( \log 4 \right)^3 , & x = 0\end{cases}\]may be continuous at x = 0 is
Options
1
2
3
none of these
Advertisements
Solution
none of these
For f(x) to be continuous at \[x = 0\] , we must have
\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]
\[\Rightarrow \lim_{x \to 0} \left[ \frac{\frac{\left( 4^x - 1 \right)^3}{x^3}}{\frac{\sin\frac{x}{a}\log\left( 1 + \frac{x^2}{3} \right)}{x^3}} \right] = 12 \left( \log 4 \right)^3 \]
\[ \Rightarrow \lim_{x \to 0} \left[ \frac{a \left( \frac{4^x - 1}{x} \right)^3}{\left( \frac{\sin\frac{x}{a}}{\frac{x}{a}} \right)\frac{\log\left( 1 + \frac{x^2}{3} \right)}{x^2}} \right] = 12 \left( \log 4 \right)^3 \]
\[ \Rightarrow 3a \lim_{x \to 0} \left[ \frac{\left( \frac{4^x - 1}{x} \right)^3}{\left( \frac{\sin\frac{x}{a}}{\frac{x}{a}} \right)\frac{\log\left( 1 + \frac{x^2}{3} \right)}{\left( \frac{x^2}{3} \right)}} \right] = 12 \left( \log 4 \right)^3 \]
\[ \Rightarrow 3a\left[ \frac{\lim_{x \to 0} \left( \frac{4^x - 1}{x} \right)^3}{\lim_{x \to 0} \left( \frac{\sin\frac{x}{a}}{\frac{x}{a}} \right) \lim_{x \to 0} \frac{\log\left( 1 + \frac{x^2}{3} \right)}{\left( \frac{x^2}{3} \right)}} \right] = 12 \left( \log 4 \right)^3 \]
\[ \Rightarrow 3a \left( \log 4 \right)^3 = 12 \left( \log 4 \right)^3 \left[ \because \lim_{x \to 0} \left( \frac{a^x - 1}{x} \right) = \log a, \lim_{x \to 0} \frac{\log\left( 1 + x \right)}{x} = 1 and \lim_{x \to 0} \frac{\sin x}{x} = 1 \right]\]
\[ \Rightarrow a = 4\]
APPEARS IN
RELATED QUESTIONS
A function f (x) is defined as
f (x) = x + a, x < 0
= x, 0 ≤x ≤ 1
= b- x, x ≥1
is continuous in its domain.
Find a + b.
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx +1", if" x<= pi),(cos x", if" x > pi):}` at x = π
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx + 1", if" x <= 5),(3x - 5", if" x > 5):}` at x = 5
Extend the definition of the following by continuity
If \[f\left( x \right) = \frac{2x + 3\ \text{ sin }x}{3x + 2\ \text{ sin } x}, x \neq 0\] If f(x) is continuous at x = 0, then find f (0).
Find the values of a and b so that the function f given by \[f\left( x \right) = \begin{cases}1 , & \text{ if } x \leq 3 \\ ax + b , & \text{ if } 3 < x < 5 \\ 7 , & \text{ if } x \geq 5\end{cases}\] is continuous at x = 3 and x = 5.
If \[f\left( x \right) = \begin{cases}\frac{x^2}{2}, & \text{ if } 0 \leq x \leq 1 \\ 2 x^2 - 3x + \frac{3}{2}, & \text P{ \text{ if } } 1 < x \leq 2\end{cases}\]. Show that f is continuous at x = 1.
If \[f\left( x \right) = \begin{cases}2 x^2 + k, &\text{ if } x \geq 0 \\ - 2 x^2 + k, & \text{ if } x < 0\end{cases}\] then what should be the value of k so that f(x) is continuous at x = 0.
Prove that the function \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & x < 0 \\ x + 1, & x \geq 0\end{cases}\] is everywhere continuous.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 - 16}{x - 2}, & \text{ if } x \neq 2 \\ 16 , & \text{ if } x = 2\end{cases}\]
The function \[f\left( x \right) = \begin{cases}x^2 /a , & \text{ if } 0 \leq x < 1 \\ a , & \text{ if } 1 \leq x < \sqrt{2} \\ \frac{2 b^2 - 4b}{x^2}, & \text{ if } \sqrt{2} \leq x < \infty\end{cases}\] is continuous on (0, ∞), then find the most suitable values of a and b.
The function f(x) is defined as follows:
If f is continuous on [0, 8], find the values of a and b.
Show that the function g (x) = x − [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.
Show that f (x) = cos x2 is a continuous function.
Show that f (x) = | cos x | is a continuous function.
Determine the value of the constant 'k' so that function f
If f (x) = (x + 1)cot x be continuous at x = 0, then f (0) is equal to
If f is defined by \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\]
If \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of
If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is
The function f (x) = x − [x], where [⋅] denotes the greatest integer function is
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
The function f(x) = `"e"^|x|` is ______.
If f.g is continuous at x = a, then f and g are separately continuous at x = a.
`lim_("x" -> 0) (1 - "cos" 4 "x")/"x"^2` is equal to ____________.
`lim_("x" -> 0) (1 - "cos x")/"x sin x"` is equal to ____________.
If `f(x) = {{:(-x^2",", "when" x ≤ 0),(5x - 4",", "when" 0 < x ≤ 1),(4x^2 - 3x",", "when" 1 < x < 2),(3x + 4",", "when" x ≥ 2):}`, then
Let f(x) = `{{:(5^(1/x), x < 0),(lambda[x], x ≥ 0):}` and λ ∈ R, then at x = 0
The function f(x) = 5x – 3 is continuous at x =
For what value of `k` the following function is continuous at the indicated point
`f(x) = {{:(kx + 1",", if x ≤ pi),(cos x",", if x > pi):}` at = `pi`
The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",", if x ≠ 0),(k",", if x = 0):}` is continuous at x = 0 is ______.
Discuss the continuity of the following function:
f(x) = sin x – cos x
For \[\frac{f}{g}\] to be continuous at \[x=c\], which condition is required?
If \[g\] is continuous and \[g(x)\neq 0\], which function is continuous?
At which values is \[\tan x=\dfrac{\sin x}{\cos x}\] not continuous?
Why is \[f(x)=\sin(x^2)\] continuous for all real \[x\]?
