Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Examine the following function for continuity:
f(x) = | x – 5|
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.
Show that
is discontinuous 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
If \[f\left( x \right) = \begin{cases}\frac{x^2 - 16}{x - 4}, & \text{ if } x \neq 4 \\ k , & \text{ if } x = 4\end{cases}\] is continuous at x = 4, find k.
If \[f\left( x \right) = \left\{ \begin{array}a x^2 + b , & 0 \leq x < 1 \\ 4 , & x = 1 \\ x + 3 , & 1 < x \leq 2\end{array}, \right.\] then the value of (a, b) for which f (x) cannot be continuous at x = 1, is
The values of the constants a, b and c for which the function \[f\left( x \right) = \begin{cases}\left( 1 + ax \right)^{1/x} , & x < 0 \\ b , & x = 0 \\ \frac{\left( x + c \right)^{1/3} - 1}{\left( x + 1 \right)^{1/2} - 1}, & x > 0\end{cases}\] may be continuous at x = 0, are
Discuss the continuity and differentiability of
If \[f\left( x \right) = \left| \log_e |x| \right|\]
Discuss continuity of f(x) =`(x^3-64)/(sqrt(x^2+9)-5)` For x ≠ 4
= 10 for x = 4 at x = 4
Find `dy/dx if y = tan^-1 ((6x)/[ 1 - 5x^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
The function f(x) = |x| + |x – 1| is ______.
f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if" x ≠ 2),(5",", "if" x = 2):}` at x = 2
f(x) = `{{:(|x|cos 1/x",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
f(x) = `{{:(x^2/2",", "if" 0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",", "if" 1 < x ≤ 2):}` at x = 1
Given the function f(x) = `1/(x + 2)`. Find the points of discontinuity of the composite function y = f(f(x))
The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.
