Advertisements
Advertisements
Question
Show that the function f defined by f(x) = `{{:(x sin 1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.
Advertisements
Solution
Left hand limit at x = 0 is given by
`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) x sin 1/x` = 0 ....`["since", -1 < sin 1/x < 1]`
Similarly `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) x sin 1/x` = 0.
. Moreover f(0) = 0.
Thus `lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) "f"(x)`
= f(0)
Hence f is continuous at x = 0.
APPEARS IN
RELATED QUESTIONS
Determine the value of 'k' for which the following function is continuous at x = 3
`f(x) = {(((x + 3)^2 - 36)/(x - 3), x != 3), (k, x = 3):}`
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(2x", if" x < 0),(0", if" 0 <= x <= 1),(4x", if" x > 1):}`
If \[f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}\]
Find whether f(x) is continuous at x = 0.
Show that
Discuss the continuity of the function f(x) at the point x = 1/2, where \[f\left( x \right) = \begin{cases}x, 0 \leq x < \frac{1}{2} \\ \frac{1}{2}, x = \frac{1}{2} \\ 1 - x, \frac{1}{2} < x \leq 1\end{cases}\]
Determine the value of the constant k so that the function
\[f\left( x \right) = \begin{cases}k x^2 , if & x \leq 2 \\ 3 , if & x > 2\end{cases}\text{is continuous at x} = 2 .\]
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.
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}k( x^2 + 2), \text{if} & x \leq 0 \\ 3x + 1 , \text{if} & x > 0\end{cases}\]
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\]
Find f (0), so that \[f\left( x \right) = \frac{x}{1 - \sqrt{1 - x}}\] becomes continuous at x = 0.
If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then
The function \[f\left( x \right) = \begin{cases}\frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ 0 , & x = 0\end{cases}\]
If \[f\left( x \right) = \begin{cases}a \sin\frac{\pi}{2}\left( x + 1 \right), & x \leq 0 \\ \frac{\tan x - \sin x}{x^3}, & x > 0\end{cases}\] is continuous at x = 0, then a equals
Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.
Show that f(x) = x1/3 is not differentiable at x = 0.
Show that the function
\[f\left( x \right) = \begin{cases}\left| 2x - 3 \right| \left[ x \right], & x \geq 1 \\ \sin \left( \frac{\pi x}{2} \right), & x < 1\end{cases}\] is continuous but not differentiable at x = 1.
If \[f\left( x \right) = \begin{cases}a x^2 - b, & \text { if }\left| x \right| < 1 \\ \frac{1}{\left| x \right|} , & \text { if }\left| x \right| \geq 1\end{cases}\] is differentiable at x = 1, find a, b.
Discuss the continuity and differentiability of f (x) = |log |x||.
Is every differentiable function continuous?
If f (x) is differentiable at x = c, then write the value of
If \[f\left( x \right) = x^2 + \frac{x^2}{1 + x^2} + \frac{x^2}{\left( 1 + x^2 \right)} + . . . + \frac{x^2}{\left( 1 + x^2 \right)} + . . . . ,\]
then at x = 0, f (x)
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
Evaluate :`int Sinx/(sqrt(cos^2 x-2 cos x-3)) dx`
If f is continuous at x = 0, then find f (0).
Where f(x) = `(3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0`
If the function f is continuous at = 2, then find f(2) where f(x) = `(x^5 - 32)/(x - 2)`, for ≠ 2.
If f(x) = `(e^(2x) - 1)/(ax)` . for x < 0 , a ≠ 0
= 1. for x = 0
= `(log(1 + 7x))/(bx)`. for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b
Examine the continuity of the following function :
`{:(,f(x),=(x^2-16)/(x-4),",","for "x!=4),(,,=8,",","for "x=4):}} " at " x=4`
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
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 f(x) = `{{:((x^3 + x^2 - 16x + 20)/(x - 2)^2",", x ≠ 2),("k"",", x = 2):}` is continuous at x = 2, find the value of k.
The number of points at which the function f(x) = `1/(log|x|)` is discontinuous is ______.
f(x) = `{{:((1 - cos 2x)/x^2",", "if" x ≠ 0),(5",", "if" x = 0):}` at x = 0
f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if" x ≠ 2),(5",", "if" x = 2):}` at x = 2
f(x) = `{{:((1 - cos "k"x)/(xsinx)",", "if" x ≠ 0),(1/2",", "if" x = 0):}` at x = 0
