Advertisements
Advertisements
Question
Discuss the continuity of the following functions. If the function have a removable discontinuity, redefine the function so as to remove the discontinuity
`f(x)=(4^x-e^x)/(6^x-1)` for x ≠ 0
`=log(2/3) ` for x=0
Solution
f(0) = log `(2/3)` .....Given .....(1)
`lim_(x->0^-f(x))=lim_(x->0^-)(4^x-e^x)/(6^x-1)`
`= lim_(x->0) ((4^x - 1) - (e^x - 1))/(6^x - 1) `
`lim_(x->0) ((4^x - 1)/x - (e^x - 1)/x)/((6^x - 1)/x)` .....[x → 0 , x ≠ 0]
`lim_(x->0) (lim_(x->0)(4^x - 1)/x - lim_(x->0)(e^x - 1)/x)/(lim_(x->0)(6^x - 1)/x)`
`=((log4) - 1)/log6` .......`[because lim_(x->0) (a^x - 1)/x = log e]`
`therefore lim_(x->0) f(x) = ((log4) - loge)/log6`
`therefore lim_( x ->0) = [log4]/[loge.log6]`
`therefore lim_( x ->0) = [log4]/[1.log6]`
`therefore lim_( x ->0) = log(2/3)`
From (1) and (2) , lim_(x->0) f(x) ≠ f(0)
∴ f is discontinuous at x = 0
Here lim_(x->0) f(x) exists but not equal to f(0). Hence , the discontinuity at x = 0 is removable and can be removed by redefining the function as follows :
`f(x)=(4^x-e^x)/(6^x-1)` for x ≠ 0
`=log(2/3) ` for x=0
APPEARS IN
RELATED QUESTIONS
Show that the function `f(x)=|x-3|,x in R` is continuous but not differentiable at x = 3.
Prove that the function `f(x) = x^n` is continuous at x = n, where n is a positive integer.
Is the function f defined by f(x)= `{(x, if x<=1),(5, if x > 1):}` continuous at x = 0? At x = 1? At x = 2?
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/|x|, if x<0),(-1, if x >= 0):}`
Is the function defined by `f(x) = {(x+5, if x <= 1),(x -5, if x > 1):}` a continuous function?
Show that the function defined by g(x) = x = [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.
Find all the points of discontinuity of f defined by `f(x) = |x| - |x + 1|`.
Using mathematical induction prove that `d/(dx) (x^n) = nx^(n -1)` for all positive integers n.
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
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: \[f\left( x \right) = \begin{cases}2x , & \text{ if } & x < 0 \\ 0 , & \text{ if } & 0 \leq x \leq 1 \\ 4x , & \text{ if } & x > 1\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}x^{10} - 1, & \text{ if } x \leq 1 \\ x^2 , & \text{ if } x > 1\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou:
Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.
Prove that `1/2 "cos"^(-1) ((1-"x")/(1+"x")) = "tan"^-1 sqrt"x"`
Show that the function f given by:
`f(x)={((e^(1/x)-1)/(e^(1/x)+1),"if",x,!=,0),(-1,"if",x,=,0):}"`
is discontinuous at x = 0.
If f(x) = `{{:("a"x + 1, "if" x ≥ 1),(x + 2, "if" x < 1):}` is continuous, then a should be equal to ______.
`lim_("x"-> 0) sqrt(1/2 (1 - "cos" 2"x"))/"x"` is equal to
The function `f(x) = (x^2 - 25)/(x + 5)` is continuous at x =
The point of discountinuity of the function `f(x) = {{:(2x + 3",", x ≤ 2),(2x - 3",", x > 2):}` is are
How many point of discontinuity for the following function in its. domain.
`f(x) = {{:(x/|x|",", if x < 0),(-1",", if x ≥ 0):}`
How many point of discontinuity for the following function for x ∈ R
`f(x) = {{:(x + 1",", if x ≥ 1),(x^2 + 1",", if x < 1):}`
`f(x) = {{:(x^10 - 1",", if x ≤ 1),(x^2",", if x > 1):}` is discontinuous at
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) = `{{:((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 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?
