Advertisements
Advertisements
प्रश्न
Discuss continuity of f(x) =`(x^3-64)/(sqrt(x^2+9)-5)` For x ≠ 4
= 10 for x = 4 at x = 4
उत्तर
`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
संबंधित प्रश्न
Examine the following function for continuity:
f (x) = x – 5
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.
For what value of k is the following function continuous at x = 2?
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].
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 \[f\left( x \right) = \begin{cases}\frac{\sin (a + 1) x + \sin x}{x} , & x < 0 \\ c , & x = 0 \\ \frac{\sqrt{x + b x^2} - \sqrt{x}}{bx\sqrt{x}} , & x > 0\end{cases}\]is continuous at x = 0, then
Let f (x) = |x| and g (x) = |x3|, then
The set of points where the function f (x) = x |x| is differentiable is
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`
`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?
If the function f is continuous at x = 0
Where f(x) = 2`sqrt(x^3 + 1)` + a, for x < 0,
= `x^3 + a + b, for x > 0
and f (1) = 2, then find a and b.
The total cost C for producing x units is Rs (x2 + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.
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 set of points where the functions f given by f(x) = |x – 3| cosx is differentiable is ______.
f(x) = `{{:(|x|cos 1/x",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if" x ≠ 0),(0",", "if" x = 0):}` at x = 0
f(x) = |x| + |x − 1| at x = 1
Find the values of a and b such that the function f defined by
f(x) = `{{:((x - 4)/(|x - 4|) + "a"",", "if" x < 4),("a" + "b"",", "if" x = 4),((x - 4)/(|x - 4|) + "b"",", "if" x > 4):}`
is a continuous function at x = 4.
Write the number of points where f(x) = |x + 2| + |x - 3| is not differentiable.
