Advertisements
Advertisements
Question
Let \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}, x \neq \frac{\pi}{4} .\] The value which should be assigned to f (x) at \[x = \frac{\pi}{4},\]so that it is continuous everywhere is
Options
1
1/2
2
none of these
Advertisements
Solution
\[\frac{1}{2}\]
If \[f\left( x \right)\] is continuous at \[x = \frac{\pi}{4}\]
If \[\frac{\pi}{4} - x = y\], then
\[x \to \frac{\pi}{4} \text{ and } y \to 0\]
\[\therefore \lim_{y \to 0} \left( \frac{\tan y}{\cot 2\left( \frac{\pi}{4} - y \right)} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \lim_{y \to 0} \left( \frac{\tan y}{\cot\left( \frac{\pi}{2} - 2y \right)} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \lim_{y \to 0} \left( \frac{\tan y}{\tan 2y} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \lim_{y \to 0} \left( \frac{\frac{\tan y}{y}}{\frac{\tan 2y}{y}} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \lim_{y \to 0} \left( \frac{\frac{\tan y}{y}}{\frac{2 \tan 2y}{2y}} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \frac{1}{2} \lim_{y \to 0} \left( \frac{\frac{\tan y}{y}}{\frac{\tan 2y}{2y}} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \frac{1}{2}\left( \frac{\lim_{y \to 0} \frac{\tan y}{y}}{\lim_{y \to 0} \frac{\tan 2y}{2y}} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow \frac{1}{2}\left( \frac{1}{1} \right) = f\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow f\left( \frac{\pi}{4} \right) = \frac{1}{2}\]
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.
If f (x) is continuous on [–4, 2] defined as
f (x) = 6b – 3ax, for -4 ≤ x < –2
= 4x + 1, for –2 ≤ x ≤ 2
Show that a + b =`-7/6`
Find the relationship between a and b so that the function f defined by f(x) = `{(ax + 1", if" x<= 3),(bx + 3", if" x > 3):}` is continuous at x = 3.
For what value of λ is the function defined by f(x) = `{(λ(x^2 - 2x)", if" x <= 0),(4x+ 1", if" x > 0):}` continuous at x = 0? What about continuity at x = 1?
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx^2", if" x<= 2),(3", if" x > 2):}` at x = 2
Show that the function defined by f(x) = cos (x2) is a continuous function.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}(x - 1)\tan\frac{\pi x}{2}, \text{ if } & x \neq 1 \\ k , if & x = 1\end{cases}\] at x = 1at x = 1
Discuss the continuity of the function
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}kx + 5, & \text{ if } x \leq 2 \\ x - 1, & \text{ if } x > 2\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}4 , & \text{ if } x \leq - 1 \\ a x^2 + b, & \text{ if } - 1 < x < 0 \\ \cos x, &\text{ if }x \geq 0\end{cases}\]
The function f(x) is defined as follows:
If f is continuous on [0, 8], find the values of a and b.
Discuss the continuity of the following functions:
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x cos x
Show that f (x) = cos x2 is a continuous function.
Show that f (x) = | cos x | is a continuous function.
If \[f\left( x \right) = \begin{cases}\frac{x}{\sin 3x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then write the value of k.
Determine whether \[f\left( x \right) = \binom{\frac{\sin x^2}{x}, x \neq 0}{0, x = 0}\] is continuous at x = 0 or not.
If \[f\left( x \right) = \binom{\frac{1 - \cos x}{x^2}, x \neq 0}{k, x = 0}\] is continuous at x = 0, find k.
\[f\left( x \right) = \begin{cases}\frac{\left| x^2 - x \right|}{x^2 - x}, & x \neq 0, 1 \\ 1 , & x = 0 \\ - 1 , & x = 1\end{cases}\] then f (x) is continuous for all
If \[f\left( x \right) = \begin{cases}\frac{\log\left( 1 + ax \right) - \log\left( 1 - bx \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] and f (x) is continuous at x = 0, then the value of k is
\[f\left( x \right) = \frac{\left( 27 - 2x \right)^{1/3} - 3}{9 - 3 \left( 243 + 5x \right)^{1/5}}\left( x \neq 0 \right)\] is continuous, is given by
The function
If \[f\left( x \right) = x \sin\frac{1}{x}, x \neq 0,\]then the value of the function at x = 0, so that the function is continuous at x = 0, is
Find the values of a and b, if the function f defined by
If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
The function \[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\] , where [⋅] denotes the greatest integer function, is
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
Let f(x) = |sin x|. Then ______.
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 ____________.
The value of f(0) for the function `f(x) = 1/x[log(1 + x) - log(1 - x)]` to be continuous at x = 0 should be
If `f`: R → {0, 1} is a continuous surjection map then `f^(-1) (0) ∩ f^(-1) (1)` is:
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 =
The function f(x) = x2 – sin x + 5 is continuous at x =
The function f(x) = x |x| is ______.
