Advertisements
Advertisements
Question
Discuss the continuity of the following function, at x = 0.
`f(x)=x/|x|,for x ne0`
`=1,`for `x=0`
Solution
f(0)=1............(given)............(1)
for` x>0, |x|=x`
`lim_(x->0^+)f(x)=lim_(x->0^+)x/|x|`
`=lim_(x->0^+)x/x`
`=lim_(x->0^+)(1)`
`=1`
for x<0,|x|=-x
`lim_(x->0^-)f(x)=lim_(x->0^+)x/|x|`
`=lim_(x->0^-)-x/x`
`=lim_(x->0^-)(-1)`
`=-1`
`therefore lim_(x->0^+)f(x)nelim_(x->0^-)f(x)`
f is discontinuous at x = 0
here `lim_(x->0^+)f(x)nelim_(x->0^-)f(X)`
`therefore lim_(x->0)f(x) `does not exist
hence, it is discontinuous at x = 0
APPEARS IN
RELATED QUESTIONS
if the function
`f(x)=k+x, for x<1`
`=4x+3, for x>=1`
id continuous at x=1 then k=
(a) 7
(b) 8
(c) 6
(d) -6
If `f(x)=[tan(pi/4+x)]^(1/x), `
= k ,for x=0
is continuous at x=0 , find k.
Find a and b, so that the function f(x) defined by
f(x)=-2sin x, for -π≤ x ≤ -π/2
=a sin x+b, for -π/2≤ x ≤ π/2
=cos x, for π/2≤ x ≤ π
is continuous on [- π, π]
If f (x) = x 2 + a, for x ≥ 0 ` =2sqrt(x^2+1)+b, ` is continuous at x = 0, find a and b.
If `f(x) =(e^(x^2)-cosx)/x^2`, for x= 0, is continuous at x = 0, find f(0).
If the function `f(x)=(5^sinx-1)^2/(xlog(1+2x))` for x ≠ 0 is continuous at x = 0, find f (0).
Examine continuity of the function f(x) at x = 0, where
`f(x) = (10^x + 7^x - 14^x - 5^x)/(1-cos 4x) , " for " x != 0`
`= 10/7 , " for" x = 0`
Discuss the continuity of the function
f(x) = `("log"(2+"x") - "log"(2-"x"))/("tan""x")` , for x ≠ 0
= 1 for x = 0 at the point x =0
`lim_(x->1) (x^2 - 3x + 2)/(x^2 - 1)` = ?
The value of p and q for which the function
f(x) = `{((sin ("p" + 1)x + sin x)/x"," x < 0),(q "," x = 0),((sqrt(x + x^2) - sqrtx)/(x^(3//2))"," x > 0):}`
is continuous for all x in R, are
If f(x) = `(1 - sin x + cos x)/(1 + sin x + cos x)`, for x ≠ π is continuous at x = π, then t(π) = ______.
If f(x) = `(sin(x^2 - x))/x`, x ≠ 0, is continuous at x = 0, then f(0) is ______
If f(x) = `{{:((cosec^2x)^{tan^2x}"," x ≠ 0), (k"," x = 0):}` is continuous at x = 0, then k is equal to ______
If f(x) = `{((3^x - e^x)/(tan2x); x ≠ 0),(1/2(log3 + 1); x = 0):}` then ______
The function `tanx/|x|` ______
lf f(x) = `(20^x + 3^x - 6^x - 10^x)/(1 - cos8x)`, for x ≠ 0
= `(k/16)log(10/3).log2`, for x = 0
is continuous at x = 0, then the value of k is ______
If f(x) = `{{:((sqrt(1 - "ax") - sqrt(1 + "ax"))/"x""," -1 ≤ "x" ≤ 0), ((3x + 1)/(x - 2)"," 0 ≤ x ≤ 1):}` is continuous in [-1, 1], then a is equal to ______
Let f(x) = `{{:((tan pix)/(7x)";", x ≠ 0),("k"";", x = 0):}`. If f(x) is continuous at x = 0, then k = ______.
If f(x) = `{{:(4x - 2",", 0 ≤ x ≤ 2),(3x + "k"",", 2 < x ≤ 3):}` is continuous in [0, 3], then the value of k is ______.
If f(x) = x2 + α for x > 0 = `2sqrt(x^2 + 1)` + β for x < 0 is continuous at x = 0 and `f(1/2)` = 2 then α2 + β2 is ______.
If f(x) = `(e^(x^2) - cos x)/x^2`, for x ≠ 0 is continuous at x = 0, then value of (0) is ______.
