Advertisements
Advertisements
प्रश्न
Test the continuity of the function on f(x) at the origin:
\[f\left( x \right) = \begin{cases}\frac{x}{\left| x \right|}, & x \neq 0 \\ 1 , & x = 0\end{cases}\]
उत्तर
Given:
We observe
(LHL at x = 0) =
(RHL at x = 0) =
Hence
APPEARS IN
संबंधित प्रश्न
Show that the function `f(x)=|x-3|,x in R` is continuous but not differentiable at x = 3.
Find the values of p and q for which
f(x) = `{((1-sin^3x)/(3cos^2x),`
is continuous at x = π/2.
Examine the continuity of the function f(x) = 2x2 – 1 at x = 3.
Prove that the function `f(x) = x^n` is continuous at x = n, where n is a positive integer.
Find all point of discontinuity of f, where f is defined by `f (x) = {(2x + 3, if x<=2),(2x - 3, if x > 2):}`
Find all points of discontinuity of f, where f is defined by `f(x) = {(|x|+3, if x<= -3),(-2x, if -3 < x < 3),(6x + 2, if x >= 3):}`
Find all points of discontinuity of f, where f is defined by `f (x) = {(x+1, if x>=1),(x^2+1, if x < 1):}`
Find all points of discontinuity of f, where f is defined by `f (x) = {(x^10 - 1, if x<=1),(x^2, if x > 1):}`
Is the function defined by `f(x) = {(x+5, if x <= 1),(x -5, if x > 1):}` a continuous function?
Determine if f defined by `f(x) = {(x^2 sin 1/x, "," if x != 0),(0, "," if x = 0):}` is a continuous function?
Using mathematical induction prove that `d/(dx) (x^n) = nx^(n -1)` for all positive integers n.
Find the value of constant ‘k’ so that the function f (x) defined as
f(x) = `{((x^2 -2x-3)/(x+1), x != -1),(k, x != -1):}`
is continous at x = -1
Find the point of discontinuity, if any, of the following function: \[f\left( x \right) = \begin{cases}\sin x - \cos x , & \text{ if } x \neq 0 \\ - 1 , & \text{ if } x = 0\end{cases}\]
Prove that `1/2 "cos"^(-1) ((1-"x")/(1+"x")) = "tan"^-1 sqrt"x"`
If f(x) = `{{:("a"x + 1, "if" x ≥ 1),(x + 2, "if" x < 1):}` is continuous, then a should be equal to ______.
Find all points of discontinuity of the function f(t) = `1/("t"^2 + "t" - 2)`, where t = `1/(x - 1)`
The number of discontinuous functions y(x) on [-2, 2] satisfying x2 + y2 = 4 is ____________.
Let f (x) `= (1 - "tan x")/(4"x" - pi), "x" ne pi/4, "x" in (0, pi/2).` If f(x) is continuous in `(0, pi/2), "then f"(pi/4) =` ____________.
The domain of the function f(x) = `""^(24 - x)C_(3x - 1) + ""^(40 - 6x)C_(8x - 10)` is
Let a, b ∈ R, b ≠ 0. Define a function
F(x) = `{{:(asin π/2(x - 1)",", "for" x ≤ 0),((tan2x - sin2x)/(bx^3)",", "for" x > 0):}`
If f is continuous at x = 0, then 10 – ab is equal to ______.
If function f(x) = `{{:((asinx + btanx - 3x)/x^3,",", x ≠ 0),(0,",", x = 0):}` is continuous at x = 0 then (a2 + b2) is equal to ______.
If functions g and h are defined as
g(x) = `{{:(x^2 + 1, x∈Q),(px^2, x\cancel(∈)Q):}`
and h(x) = `{{:(px, x∈Q),(2x + q, x\cancel(∈)Q):}`
If (g + h)(x) is continuous at x = 1 and x = 3, then 3p + q is ______.
If f(x) = `{{:(cos ((π(sqrt(1 + x) - 1))/x)/x,",", x ≠ 0),(π/k,",", x = 0):}`
is continuous at x = 0, then k2 is equal to ______.
If f(x) = `{{:((log_(sin|x|) cos^2x)/(log_(sin|3x|) cos x/2), |x| < π/3; x ≠ 0),(k, x = 0):}`, then value of k for which f(x) is continuous at x = 0 is ______.
If the function f defined as f(x) = `1/x - (k - 1)/(e^(2x) - 1)` x ≠ 0, is continuous at x = 0, then the ordered pair (k, f(0)) is equal to ______.
Find the value(s) of 'λ' if the function
f(x) = `{{:((sin^2 λx)/x^2",", if x ≠ 0 "is continuous at" x = 0.),(1",", if x = 0):}`
Find the value of k for which the function f given as
f(x) =`{{:((1 - cosx)/(2x^2)",", if x ≠ 0),( k",", if x = 0 ):}`
is continuous at x = 0.
The graph of the function f is shown below.
Of the following options, at what values of x is the function f NOT differentiable?
Consider the graph `y = x^(1/3)`
Statement 1: The above graph is continuous at x = 0
Statement 2: The above graph is differentiable at x = 0
Which of the following is correct?
