Advertisements
Advertisements
प्रश्न
Discuss the continuity of the function `f(x) = (3 - sqrt(2x + 7))/(x - 1)` for x ≠ 1
= `-1/3` for x = 1, at x = 1
उत्तर
Given `f(1) = (-1)/3`
Consider
`lim_(x ->1) f(x) = lim_(x ->1) (3 - sqrt(2x + 7))/(x - 1)`
= `lim_(x ->1) (3 - sqrt(2x + 7))/(x - 1) xx (3 + sqrt(2x + 7))/(3 + sqrt(2x + 7))`
= `lim_(x ->1) (9 - 2x - 7)/((x - 1) (3 + sqrt(2x + 7))`
= `lim_(x ->1) (2 - 2x)/((x - 1) (3 + sqrt(2x + 7))`
= `lim_(x ->1) (-2(x - 1))/((x - 1) (3 + sqrt(2x + 7))`
= `lim_(x ->1) (-2)/(3 + sqrt(2x + 7)`
(`therefore x -> 1, ( x- 1) ≠ 0)`
= `(-2)/(3 + sqrt(2x + 7)`
= `(-2)/(3 + sqrt9)`
= `(-2)/(3 + 3)`
= `(-2)/6`
= `(-1)/3`
`lim_(x ->1) f(x) = f(1)`
Function is continuous at x = 1
APPEARS IN
संबंधित प्रश्न
Examine the following function for continuity:
f (x) = x – 5
Discuss the continuity of the function f, where f is defined by `f(x) = {(-2,"," if x <= -1),(2x, "," if -1 < x <= 1),(2, "," if x > 1):}`
A function f(x) is defined as
Show that f(x) is continuous at x = 3
Let \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x^2}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}\] Show that f(x) is discontinuous at x = 0.
Discuss the continuity of the following functions at the indicated point(s): (iv) \[f\left( x \right) = \left\{ \begin{array}{l}\frac{e^x - 1}{\log(1 + 2x)}, if & x \neq a \\ 7 , if & x = 0\end{array}at x = 0 \right.\]
Show that
\[f\left( x \right) = \begin{cases}1 + x^2 , if & 0 \leq x \leq 1 \\ 2 - x , if & x > 1\end{cases}\]
Determine the values of a, b, c for which the function f(x) = `{((sin(a + 1)x + sin x)/x, "for" x < 0),(x, "for" x = 0),((sqrt(x + bx^2) - sqrtx)/(bx^(3"/"2)), "for" x > 0):}` is continuous at x = 0.
Discuss the continuity of the function \[f\left( x \right) = \begin{cases}2x - 1 , & \text { if } x < 2 \\ \frac{3x}{2} , & \text{ if } x \geq 2\end{cases}\]
If \[f\left( x \right) = \begin{cases}\frac{\sin (a + 1) x + \sin x}{x} , & x < 0 \\ c , & x = 0 \\ \frac{\sqrt{x + b x^2} - \sqrt{x}}{bx\sqrt{x}} , & x > 0\end{cases}\]is continuous at x = 0, then
The points of discontinuity of the function\[f\left( x \right) = \begin{cases}\frac{1}{5}\left( 2 x^2 + 3 \right) , & x \leq 1 \\ 6 - 5x , & 1 < x < 3 \\ x - 3 , & x \geq 3\end{cases}\text{ is } \left( are \right)\]
The function f (x) = sin−1 (cos x) is
The function f (x) = e−|x| is
If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
If the function f is continuous at = 2, then find f(2) where f(x) = `(x^5 - 32)/(x - 2)`, for ≠ 2.
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
The function given by f (x) = tanx is discontinuous on the set ______.
The set of points where the functions f given by f(x) = |x – 3| cosx is differentiable is ______.
f(x) = `{{:((2^(x + 2) - 16)/(4^x - 16)",", "if" x ≠ 2),("k"",", "if" x = 2):}` at x = 2
f(x) = `{{:((1 - cos "k"x)/(xsinx)",", "if" x ≠ 0),(1/2",", "if" x = 0):}` at x = 0
