Advertisements
Advertisements
प्रश्न
If f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4` find the value of `"f"(pi/4)` so that f (x) becomes continuous at x = `pi/4`
Advertisements
उत्तर
Given, f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4`
Therefore, `lim_(x -> pi/4) "f"(x) = lim_(x -> pi/4) (sqrt(2) cos x - 1)/(cot x - 1)`
= `lim_(x -> pi/4) ((sqrt(2) cos x - 1) sin x)/(cos x - sin x)`
= `lim_(x -> pi/4) ((sqrt(2) cos x - 1))/((sqrt(2) cos x + 1)) * ((sqrt(2) cos x + 10))/((cosx - sin x)) * ((cosx + sin x))/((cos x + sin x)) * sin x`
= `lim_(x -> pi/4) (2cos^2 x - 1)/(cos^2 x - sin^2x) * (cosx + sinx)/(sqrt(2) cos x + 1) * (sin x)`
= `lim_(x -> pi/4) (cos 2x)/(cos 2x) * ((cosx + sinx)/(sqrt(2) cos x + 1)) * (sin x)`
= `lim_(x -> pi/4) ((cosx + sin x))/(sqrt(2) cos x + 1) sinx`
= `(1/sqrt(2) (1/sqrt(2) + 1/sqrt(2)))/(sqrt(2) * 1/sqrt(2) + 1)`
= `1/2`
Thus, `lim_(x -> pi/2) "f"(x) = 1/2`
If we define `"f"(pi/4) = 1/2`, then f(x) will become continuous at x = `pi/4`.
Hence for f to be continuous at x = `pi/4`, `"f"(pi/4) = 1/2`.
APPEARS IN
संबंधित प्रश्न
Examine the continuity of the following function :
`{:(,,f(x)= x^2 -x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3`
Examine the following function for continuity:
f(x) = `(x^2 - 25)/(x + 5)`, x ≠ −5
A function f(x) is defined as
Show that f(x) is continuous at x = 3
If \[f\left( x \right) = \begin{cases}\frac{x^2 - 1}{x - 1}; for & x \neq 1 \\ 2 ; for & x = 1\end{cases}\] Find whether f(x) is continuous at x = 1.
If \[f\left( x \right) = \begin{cases}e^{1/x} , if & x \neq 0 \\ 1 , if & x = 0\end{cases}\] find whether f is continuous at x = 0.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}kx + 1, if & x \leq 5 \\ 3x - 5, if & x > 5\end{cases}\] at x = 5
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{x^2 - 25}{x - 5}, & x \neq 5 \\ k , & x = 5\end{cases}\]at x = 5
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\]
Discuss the continuity of the f(x) at the indicated points:
(i) f(x) = | x | + | x − 1 | at x = 0, 1.
Find the points of discontinuity, if any, of the following functions:
Find f (0), so that \[f\left( x \right) = \frac{x}{1 - \sqrt{1 - x}}\] becomes continuous at x = 0.
If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then
If \[f\left( x \right) = \left| \log_{10} x \right|\] then at x = 1
Discuss the continuity and differentiability of f (x) = |log |x||.
Give an example of a function which is continuos but not differentiable at at a point.
Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.
Let \[f\left( x \right) = \left( x + \left| x \right| \right) \left| x \right|\]
Let f (x) = |x| and g (x) = |x3|, then
The function f (x) = e−|x| is
If \[f\left( x \right) = x^2 + \frac{x^2}{1 + x^2} + \frac{x^2}{\left( 1 + x^2 \right)} + . . . + \frac{x^2}{\left( 1 + x^2 \right)} + . . . . ,\]
then at x = 0, f (x)
Let \[f\left( x \right) = \begin{cases}1 , & x \leq - 1 \\ \left| x \right|, & - 1 < x < 1 \\ 0 , & x \geq 1\end{cases}\] Then, f is
Examine the continuity of f(x)=`x^2-x+9 "for" x<=3`
=`4x+3 "for" x>3, "at" x=3`
Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0
= k for x = 0
is continuous at x = 0.
If f(x) = `(e^(2x) - 1)/(ax)` . for x < 0 , a ≠ 0
= 1. for x = 0
= `(log(1 + 7x))/(bx)`. for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b
Examine the continuity off at x = 1, if
f (x) = 5x - 3 , for 0 ≤ x ≤ 1
= x2 + 1 , for 1 ≤ x ≤ 2
Examine the continuity of the followin function :
`{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0`
Discuss the continuity of the function f(x) = sin x . cos x.
Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if" x ≠ 0),(0",", "if" x = 0):}` is discontinuous at x = 0.
A continuous function can have some points where limit does not exist.
f(x) = `{{:(|x|cos 1/x",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
f(x) = `{{:(|x - "a"| sin 1/(x - "a")",", "if" x ≠ 0),(0",", "if" x = "a"):}` at x = a
f(x) = `{{:(x^2/2",", "if" 0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",", "if" 1 < x ≤ 2):}` at x = 1
f(x) = `{{:((1 - cos "k"x)/(xsinx)",", "if" x ≠ 0),(1/2",", "if" x = 0):}` at x = 0
Examine the differentiability of f, where f is defined by
f(x) = `{{:(x[x]",", "if" 0 ≤ x < 2),((x - 1)x",", "if" 2 ≤ x < 3):}` at x = 2
Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.
Find the values of p and q so that f(x) = `{{:(x^2 + 3x + "p"",", "if" x ≤ 1),("q"x + 2",", "if" x > 1):}` is differentiable at x = 1
`lim_("x" -> "x" //4) ("cos x - sin x")/("x"- "x" /4)` is equal to ____________.
The value of k (k < 0) for which the function f defined as
f(x) = `{((1-cos"kx")/("x"sin"x")"," "x" ≠ 0),(1/2"," "x" = 0):}`
is continuous at x = 0 is:
