Advertisements
Advertisements
प्रश्न
Extend the definition of the following by continuity
Advertisements
उत्तर
Given:
If f(x) is continuous at x =
\[ \Rightarrow \lim_{x \to \pi} \frac{1 - \cos7\left( x - \pi \right)}{5 \left( x - \pi \right)^2} = f\left( \pi \right)\]
\[ \Rightarrow \frac{2}{5} \lim_{x \to \pi} \frac{\sin^2 \left( \frac{7\left( x - \pi \right)}{2} \right)}{\left( x - \pi \right)^2} = f\left( \pi \right)\]
\[ \Rightarrow \frac{2}{5} \times \frac{49}{4} \lim_{x \to \pi} \frac{\sin^2 \left( \frac{7\left( x - \pi \right)}{2} \right)}{\frac{49}{4} \left( x - \pi \right)^2} = f\left( \pi \right)\]
\[ \Rightarrow \frac{2}{5} \times \frac{49}{4} \lim_{x \to \pi} \frac{\sin^2 \left( \frac{7\left( x - \pi \right)}{2} \right)}{\left( \frac{7}{2}\left( x - \pi \right) \right)^2} = f\left( \pi \right)\]
\[ \Rightarrow \frac{2}{5} \times \frac{49}{4} \lim_{x \to \pi} \left[ \frac{\sin\left( \frac{7\left( x - \pi \right)}{2} \right)}{\left( \frac{7}{2}\left( x - \pi \right) \right)} \right]^2 = f\left( \pi \right)\]
\[ \Rightarrow \frac{2}{5} \times \frac{49}{4} \times 1 = f\left( \pi \right)\]
\[ \Rightarrow \frac{1}{5} \times \frac{49}{2} \times 1 = f\left( \pi \right)\]
\[ \Rightarrow \frac{49}{10} = f\left( \pi \right)\]
APPEARS IN
संबंधित प्रश्न
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`
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Show that the function defined by f(x) = cos (x2) is a continuous function.
Show that the function defined by f(x) = |cos x| is a continuous function.
Examine that sin |x| is a continuous function.
If \[f\left( x \right) = \begin{cases}\frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, find k.
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.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}kx + 5, & \text{ if } x \leq 2 \\ x - 1, & \text{ if } x > 2\end{cases}\]
The function f(x) is defined as follows:
If f is continuous on [0, 8], find the values of a and b.
What happens to a function f (x) at x = a, if
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.
If \[f\left( x \right) = \begin{cases}\frac{\sin^{- 1} x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]is continuous at x = 0, write the value of k.
Determine the value of the constant 'k' so that function f
\[f\left( x \right) = \begin{cases}\frac{\left| x^2 - x \right|}{x^2 - x}, & x \neq 0, 1 \\ 1 , & x = 0 \\ - 1 , & x = 1\end{cases}\] then f (x) is continuous for all
The function \[f\left( x \right) = \begin{cases}1 , & \left| x \right| \geq 1 & \\ \frac{1}{n^2} , & \frac{1}{n} < \left| x \right| & < \frac{1}{n - 1}, n = 2, 3, . . . \\ 0 , & x = 0 &\end{cases}\]
The function
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
If the function f (x) defined by \[f\left( x \right) = \begin{cases}\frac{\log \left( 1 + 3x \right) - \log \left( 1 - 2x \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then k =
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
If f is defined by \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\]
Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if
If f(x) = 2x and g(x) = `x^2/2 + 1`, then which of the following can be a discontinuous function ______.
The function f(x) = `"e"^|x|` is ______.
Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.
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:
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
The function f(x) = x2 – sin x + 5 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`
Discuss the continuity of the following function:
f(x) = sin x – cos x
