Advertisements
Advertisements
प्रश्न
Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0
= k for x = 0
is continuous at x = 0.
उत्तर
`Lim_(x→0)[log(1+3x)/(5x)]`
=`lim_(x->0) log (1 + 3x)/(5x)`
=`lim_(x->0) log (1 + 3x)/(5x) xx 3/5`
= `1 xx 3/5`
=`3/5`
∵ f is continuous at x=0
∴ `Lim_(x→0) f(x)=f(0)⇒ k=3/5`
APPEARS IN
संबंधित प्रश्न
Discuss the continuity of the following functions at the indicated point(s):
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
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}k( x^2 + 2), \text{if} & x \leq 0 \\ 3x + 1 , \text{if} & x > 0\end{cases}\]
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.
If the functions f(x), defined below is continuous at x = 0, find the value of k. \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2x}{2 x^2}, & x < 0 \\ k , & x = 0 \\ \frac{x}{\left| x \right|} , & x > 0\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 + x^3 + 2 x^2}{\tan^{- 1} x}, & \text{ if } x \neq 0 \\ 10 , & \text{ if } x = 0\end{cases}\]
Discuss the continuity of the function \[f\left( x \right) = \begin{cases}2x - 1 , & \text { if } x < 2 \\ \frac{3x}{2} , & \text{ if } x \geq 2\end{cases}\]
The value of f (0), so that the function
The function f (x) = sin−1 (cos x) is
The function f (x) = e−|x| is
Discuss the continuity of the function f(x) = sin x . cos x.
If f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4` find the value of `"f"(pi/4)` so that f (x) becomes continuous at x = `pi/4`
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?
For continuity, at x = a, each of `lim_(x -> "a"^+) "f"(x)` and `lim_(x -> "a"^-) "f"(x)` is equal to f(a).
Examine the continuity of the function f(x) = x3 + 2x2 – 1 at x = 1
Given the function f(x) = `1/(x + 2)`. Find the points of discontinuity of the composite function y = f(f(x))
Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.
`lim_("x" -> 0) (2 "sin x - sin" 2 "x")/"x"^3` is equal to ____________.
