Advertisements
Advertisements
प्रश्न
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = –3 and at x = 5.
Advertisements
उत्तर
The given function is:
f(x) = 5x – 3
f(0) = 5(0) – 3 = –3
`lim_(x → 0)` f(x) = 5(0) – 3 = –3
`lim_(x → 0)` f(x) = f(0)
Hence, the function is continuous at x = 0.
f(–3) = 5(–3) – 3
= –15 – 3
= –18
⇒ `lim_(x → -3)` f(x) = 5(–3) – 3
= –15 – 3
= –18
⇒ `lim_(x → -3)` f(x) = f(–3)
Hence, the function is continuous at x = –3.
f(5) = 5(5) – 3
= 25 – 3
= 22
⇒ `lim_(x → 5)` f(x)
= 5(5) – 3
= 25 – 3
= –22
⇒ `lim_(x -> 5)` f(x) = f(5)
Hence, the function is continuous at x = 5.
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.
Find all points 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/|x|", if" x<0),(-1", if" x >= 0):}`
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?
Examine the continuity of f, where f is defined by:
f(x) = `{(sin x - cos x", if" x != 0),(-1", if" x = 0):}`
Find all the points of discontinuity of f defined by f(x) = |x| − |x + 1|.
Determine the value of the constant 'k' so that function f(x) `{((kx)/|x|, ","if x < 0),(3"," , if x >= 0):}` is continuous at x = 0
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
Show that the function f(x) = `{(x^2, x<=1),(1/2, x>1):}` is continuous at x = 1 but not differentiable.
Test the continuity of the function on f(x) at the origin:
\[f\left( x \right) = \begin{cases}\frac{x}{\left| x \right|}, & x \neq 0 \\ 1 , & x = 0\end{cases}\]
For what value of λ is the function
\[f\left( x \right) = \begin{cases}\lambda( x^2 - 2x), & \text{ if } x \leq 0 \\ 4x + 1 , & \text{ if } x > 0\end{cases}\]continuous at x = 0? What about continuity at x = ± 1?
Find the relationship between 'a' and 'b' so that the function 'f' defined by
Find the points of discontinuity, if any, of the following functions:
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}2x , & \text{ if } & x < 0 \\ 0 , & \text{ if } & 0 \leq x \leq 1 \\ 4x , & \text{ if } & x > 1\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}- 2 , & \text{ if }& x \leq - 1 \\ 2x , & \text{ if } & - 1 < x < 1 \\ 2 , & \text{ if } & x \geq 1\end{cases}\]
Discuss the Continuity of the F(X) at the Indicated Points : F(X) = | X − 1 | + | X + 1 | at X = −1, 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"`
Find all points of discontinuity of the function f(t) = `1/("t"^2 + "t" - 2)`, where t = `1/(x - 1)`
`lim_("x" -> pi/2)` [sinx] is equal to ____________.
The number of discontinuous functions y(x) on [-2, 2] satisfying x2 + y2 = 4 is ____________.
`lim_("x"-> 0) sqrt(1/2 (1 - "cos" 2"x"))/"x"` is equal to
The function f defined by `f(x) = {{:(x, "if" x ≤ 1),(5, "if" x > 1):}` discontinuous at x equal to
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 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 ______.
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.
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?
