Advertisements
Advertisements
प्रश्न
If the function f (x) = `(15^x - 3^x - 5^x + 1)/(x tanx)`, x ≠ 0 is continuous at x = 0 , then find f(0).
उत्तर
Function f is continuous at x = 0
∴ `f(0) = lim_(x → 0) f(x)`
= `lim_(x → 0) (15^x - 3^x - 5^x + 1)/(x tanx)`
= `lim_(x → 0) (3^x.5^x - 3^x - 5^x + 1)/(x tanx)`
= `lim_(x → 0) (3^x(5^x - 1) - 1(5^x - 1))/(x tanx)`
= `lim_(x → 0) ((5^x - 1)(3^x - 1))/(x tanx)`
= `(lim_(x → 0) ((5^x - 1)(3^x - 1))/x^2)/(lim_(x → 0) (xtanx)/x^2`
= `((lim_(x → 0) (5^x - 1)/x) (lim_(x → 0) (3^x - 1)/x))/((lim_(x → 0) tanx/x))`
= `((log 5)(log 3))/1`
f(0) = (log5)(log3)
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) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`
A function f(x) is defined as
Show that f(x) is continuous at x = 3
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.
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.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}k x^2 , & x \geq 1 \\ 4 , & x < 1\end{cases}\]at x = 1
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:
Find all point of discontinuity of the function
The points of discontinuity of the function\[f\left( x \right) = \begin{cases}\frac{1}{5}\left( 2 x^2 + 3 \right) , & x \leq 1 \\ 6 - 5x , & 1 < x < 3 \\ x - 3 , & x \geq 3\end{cases}\text{ is } \left( are \right)\]
Show that \[f\left( x \right) =\]`{(12x, -,13, if , x≤3),(2x^2, +,5, if x,>3):}` is differentiable at x = 3. Also, find f'(3).
Find whether the function is differentiable at x = 1 and x = 2
Is every differentiable function continuous?
Let f (x) = |sin x|. Then,
If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
Discuss the continuity of f at x = 1
Where f(X) = `[ 3 - sqrt ( 2x + 7 ) / ( x - 1 )]` For x ≠ 1
= `-1/3` For x = 1
Find the value of 'k' if the function
f(x) = `(tan 7x)/(2x)`, for x ≠ 0.
= k for x = 0.
is continuous at x = 0.
For continuity, at x = a, each of `lim_(x -> "a"^+) "f"(x)` and `lim_(x -> "a"^-) "f"(x)` is equal to f(a).
A continuous function can have some points where limit does not exist.
