Advertisements
Advertisements
प्रश्न
If \[f\left( x \right) = \left| \log_e x \right|, \text { then}\]
विकल्प
\[f' \left( 1^+ \right) = 1\]
\[f' \left( 1 \right) = - 1\]
\[f' \left( 1 \right) = 1\]
\[f' \left( 1 \right) = - 1\]
Advertisements
उत्तर
(a)
`f(x) = |log_e x|, = {(-log_e x ,"for " 0< x<1),(log_e x ,"for "x ge 1):}`
\[\text{ Differentiablity at } x = 1, \]
we have ,
\[ (\text { LHD at x } = 1 ) = {lim}_{x \to 1^-} \frac{f(x) - f(1)}{x - 1}\]
\[ = {lim}_{x \to 1^-} \frac{- \log x - \log 1}{x - 1}\]
\[ = - {lim}_{x \to 1^-} \frac{\log x}{x - 1}\]
\[ \]
\[ = - {lim}_{h \to 0} \frac{\log (1 - h)}{1 - h - 1}\]
\[ = - {lim}_{h \to 0} \frac{\log (1 - h)}{- h} = - 1 \]
\[(\text { RHD at x } = 1 ) = {lim}_{x \to 1^+} \frac{f(x) - f(1)}{x - 1} \]
\[ = {lim}_{x \to 1^+} \frac{\log x - \log (1)}{x - 1}\]
\[ \]
\[ = {lim}_{h \to 0} \frac{\log (1 + h)}{x - 1} = {lim}_{h \to 0} \frac{\log (1 + h)}{h} = 1\]
APPEARS IN
संबंधित प्रश्न
If f(x)= `{((sin(a+1)x+2sinx)/x,x<0),(2,x=0),((sqrt(1+bx)-1)/x,x>0):}`
is continuous at x = 0, then find the values of a and b.
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(2x", if" x < 0),(0", if" 0 <= x <= 1),(4x", if" x > 1):}`
A function f(x) is defined as
Show that f(x) is continuous at x = 3
Show that
\[f\left( x \right) = \begin{cases}\frac{\sin 3x}{\tan 2x} , if x < 0 \\ \frac{3}{2} , if x = 0 \\ \frac{\log(1 + 3x)}{e^{2x} - 1} , if x > 0\end{cases}\text{is continuous at} x = 0\]
Discuss the continuity of the f(x) at the indicated points:
(i) f(x) = | x | + | x − 1 | at x = 0, 1.
If the functions f(x), defined below is continuous at x = 0, find the value of k. \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2x}{2 x^2}, & x < 0 \\ k , & x = 0 \\ \frac{x}{\left| x \right|} , & x > 0\end{cases}\]
Find the values of a and b so that the function f(x) defined by \[f\left( x \right) = \begin{cases}x + a\sqrt{2}\sin x , & \text{ if }0 \leq x < \pi/4 \\ 2x \cot x + b , & \text{ if } \pi/4 \leq x < \pi/2 \\ a \cos 2x - b \sin x, & \text{ if } \pi/2 \leq x \leq \pi\end{cases}\]becomes continuous on [0, π].
Find all point of discontinuity of the function
The value of f (0), so that the function
The value of b for which the function
The value of k which makes \[f\left( x \right) = \begin{cases}\sin\frac{1}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] continuous at x = 0, is
The points of discontinuity of the function
\[f\left( x \right) = \begin{cases}2\sqrt{x} , & 0 \leq x \leq 1 \\ 4 - 2x , & 1 < x < \frac{5}{2} \\ 2x - 7 , & \frac{5}{2} \leq x \leq 4\end{cases}\text{ is } \left( \text{ are }\right)\]
Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at x = 3. Also, find f'(3).
Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.
If \[f\left( x \right) = \left| \log_e |x| \right|\]
The set of points where the function f (x) given by f (x) = |x − 3| cos x is differentiable, is
Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0
= k for x = 0
is continuous at x = 0.
Discuss continuity of f(x) =`(x^3-64)/(sqrt(x^2+9)-5)` For x ≠ 4
= 10 for x = 4 at x = 4
Evaluate :`int Sinx/(sqrt(cos^2 x-2 cos x-3)) dx`
`f(x)=(x^2-9)/(x - 3)` is not defined at x = 3. what value should be assigned to f(3) for continuity of f(x) at = 3?
Find k, if the function f is continuous at x = 0, where
`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0
= k , for x = 0
If the function f is continuous at = 2, then find f(2) where f(x) = `(x^5 - 32)/(x - 2)`, for ≠ 2.
Examine the continuity of the following function :
f(x) = x2 - x + 9, for x ≤ 3
= 4x + 3, for x > 3
at x = 3.
Discuss the continuity of the function at the point given. If the function is discontinuous, then remove the discontinuity.
f (x) = `(sin^2 5x)/x^2` for x ≠ 0
= 5 for x = 0, at x = 0
If the function f is continuous at x = I, then find f(1), where f(x) = `(x^2 - 3x + 2)/(x - 1),` for x ≠ 1
If the function f is continuous at x = 2, then find 'k' where
f(x) = `(x^2 + 5)/(x - 1),` for 1< x ≤ 2
= kx + 1 , for x > 2
Discuss the continuity of the function f at x = 0, where
f(x) = `(5^x + 5^-x - 2)/(cos2x - cos6x),` for x ≠ 0
= `1/8(log 5)^2,` for x = 0
Find the value of the constant k so that the function f defined below is continuous at x = 0, where f(x) = `{{:((1 - cos4x)/(8x^2)",", x ≠ 0),("k"",", x = 0):}`
Discuss the continuity of the function f(x) = sin x . cos x.
The function given by f (x) = tanx is discontinuous on the set ______.
For continuity, at x = a, each of `lim_(x -> "a"^+) "f"(x)` and `lim_(x -> "a"^-) "f"(x)` is equal to f(a).
A continuous function can have some points where limit does not exist.
Examine the continuity of the function f(x) = x3 + 2x2 – 1 at x = 1
f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
f(x) = `{{:(3x - 8",", "if" x ≤ 5),(2"k"",", "if" x > 5):}` at x = 5
Examine the differentiability of f, where f is defined by
f(x) = `{{:(x^2 sin 1/x",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
