Advertisements
Advertisements
Question
Discuss continuity of f(x) =`(x^3-64)/(sqrt(x^2+9)-5)` For x ≠ 4
= 10 for x = 4 at x = 4
Solution
`f(4) = 10`
`lim_(x→4)f(x)=lim_(x→4)(x^3-64)/(sqrt(x^2+9)-5)`
`=lim_(x→4)(x^3-64)/(sqrt(x^2+9)-5)`
`lim_(x→4)(x^3-4^3)/(sqrt(x^2+9)-5)xx(sqrt(x^2+9)+5)/(sqrt(x^2+9)+5)`
`Lim_(x→4)((x^3-4^3)(sqrtx^2+9+5))/((sqrtx^2+9)^2-(-5)^2)`
`Lim_(x→4) ((x-4)(x^2+4x+16)(sqrt(x^2+9)+5))/(x^2+9-25)` .......`[a^3 – b^3 = (a – b) (a^2 + ab + b^2))`
`Lim_(x→4) ((x-4)(x^2+4x+16)(sqrt(x^2+9)+5))/(x^2-16)`
`Lim_(x→4) ((x-4)(x^2+4x+16)(sqrt(x^2+9)+5))/((x-4)(x+4)`)
`Lim_(x→4) ((x^2+4x+16)(sqrt(x^2+9)+5))/(x+4)`
`(((4)^2+4(4)+16)(sqrt(4^2+9)+5))/(4+4)`
`((16+16+16)(sqrt(16+9)+5))/8`
`((16+16+16)(sqrt(25)+5))/8`
`((16+16+16)(5+5))/8`
`(48x10)/8`
`f(x)lim_(x→4)=60`
`lim_(x→4) f(x)≠f(4)`
∴ f (x) is not continuous at x = 4.
APPEARS IN
RELATED QUESTIONS
Examine the following function for continuity:
`f (x)1/(x - 5), 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; \[f\left( x \right) = \begin{cases}k( x^2 + 2), \text{if} & x \leq 0 \\ 3x + 1 , \text{if} & x > 0\end{cases}\]
Prove that \[f\left( x \right) = \begin{cases}\frac{x - \left| x \right|}{x}, & x \neq 0 \\ 2 , & x = 0\end{cases}\] is discontinuous at x = 0
Let\[f\left( x \right) = \left\{ \begin{array}\frac{1 - \sin^3 x}{3 \cos^2 x} , & \text{ if } x < \frac{\pi}{2} \\ a , & \text{ if } x = \frac{\pi}{2} \\ \frac{b(1 - \sin x)}{(\pi - 2x )^2}, & \text{ if } x > \frac{\pi}{2}\end{array} . \right.\] ]If f(x) is continuous at x = \[\frac{\pi}{2}\] , find a and b.
Find the points of discontinuity, if any, of the following functions:
The function \[f\left( x \right) = \begin{cases}\frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ 0 , & x = 0\end{cases}\]
The value of f (0), so that the function
The value of f (0) so that the function
Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.
Find whether the function is differentiable at x = 1 and x = 2
If f is defined by f (x) = x2, find f'(2).
If f (x) is differentiable at x = c, then write the value of
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 Y = tan-1 `[(cos 2x - sin 2x)/(sin2x + cos 2x)]` then find `(dy)/(dx)`
Let f(x) = `{{:((1 - cos 4x)/x^2",", "if" x < 0),("a"",", "if" x = 0),(sqrt(x)/(sqrt(16) + sqrt(x) - 4)",", "if" x > 0):}`. For what value of a, f is continuous at x = 0?
The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.
f(x) = `{{:(3x + 5",", "if" x ≥ 2),(x^2",", "if" x < 2):}` at x = 2
f(x) = `{{:((1 - cos "k"x)/(xsinx)",", "if" x ≠ 0),(1/2",", "if" x = 0):}` at x = 0
A function f: R → R satisfies the equation f( x + y) = f(x) f(y) for all x, y ∈ R, f(x) ≠ 0. Suppose that the function is differentiable at x = 0 and f′(0) = 2. Prove that f′(x) = 2f(x).
