Advertisements
Advertisements
Question
If \[f\left( x \right) = \frac{2x + 3\ \text{ sin }x}{3x + 2\ \text{ sin } x}, x \neq 0\] If f(x) is continuous at x = 0, then find f (0).
Advertisements
Solution
Given:
\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]
\[ \Rightarrow \lim_{x \to 0} \frac{2x + 3\ sinx}{3x + 2\ sinx} = f\left( 0 \right)\]
\[ \Rightarrow \lim_{x \to 0} \frac{x\left( 2 + 3\frac{\ sinx}{x} \right)}{x\left( 3 + 2\frac{\ sinx}{x} \right)} = f\left( 0 \right)\]
\[ \Rightarrow \lim_{x \to 0} \frac{\left( 2 + 3\frac{\ sinx}{x} \right)}{\left( 3 + 2\frac{\ sinx}{x} \right)} = f\left( 0 \right)\]
\[ \Rightarrow \frac{\lim_{x \to 0} \left( 2 + 3\frac{\ sinx}{x} \right)}{\lim_{x \to 0} \left( 3 + 2\frac{\ sinx}{x} \right)} = f\left( 0 \right)\]
\[ \Rightarrow \frac{2 + 3 \lim_{x \to 0} \left( \frac{\ sinx}{x} \right)}{3 + 2 \lim_{x \to 0} \left( \frac{\ sinx}{x} \right)} = f\left( 0 \right)\]
\[ \Rightarrow \frac{2 + 3 \times 1}{3 + 2 \times 1} = f\left( 0 \right)\]
\[ \Rightarrow \frac{5}{5} = f\left( 0 \right)\]
\[ \Rightarrow f\left( 0 \right) = 1\]
APPEARS IN
RELATED QUESTIONS
If f (x) is continuous on [–4, 2] defined as
f (x) = 6b – 3ax, for -4 ≤ x < –2
= 4x + 1, for –2 ≤ x ≤ 2
Show that a + b =`-7/6`
Is the function defined by f(x) = x2 − sin x + 5 continuous at x = π?
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx + 1", if" x <= 5),(3x - 5", if" x > 5):}` at x = 5
If \[f\left( x \right) = \begin{cases}\frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, find k.
Extend the definition of the following by continuity
If \[f\left( x \right) = \begin{cases}2 x^2 + k, &\text{ if } x \geq 0 \\ - 2 x^2 + k, & \text{ if } x < 0\end{cases}\] then what should be the value of k so that f(x) is continuous at x = 0.
Prove that the function \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & x < 0 \\ x + 1, & x \geq 0\end{cases}\] is everywhere continuous.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]
If \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}\]
for x ≠ π/4, find the value which can be assigned to f(x) at x = π/4 so that the function f(x) becomes continuous every where in [0, π/2].
Show that the function g (x) = x − [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.
Discuss the continuity of the following functions:
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x cos x
What happens to a function f (x) at x = a, if
If \[f\left( x \right) = \binom{\frac{1 - \cos x}{x^2}, x \neq 0}{k, x = 0}\] is continuous at x = 0, find k.
If \[f\left( x \right) = \begin{cases}\frac{1 - \sin x}{\left( \pi - 2x \right)^2} . \frac{\log \sin x}{\log\left( 1 + \pi^2 - 4\pi x + 4 x^2 \right)}, & x \neq \frac{\pi}{2} \\ k , & x = \frac{\pi}{2}\end{cases}\]is continuous at x = π/2, then k =
If f (x) = (x + 1)cot x be continuous at x = 0, then f (0) is equal to
If \[f\left( x \right) = \frac{1 - \sin x}{\left( \pi - 2x \right)^2},\] when x ≠ π/2 and f (π/2) = λ, then f (x) will be continuous function at x= π/2, where λ =
If the function \[f\left( x \right) = \frac{2x - \sin^{- 1} x}{2x + \tan^{- 1} x}\] is continuous at each point of its domain, then the value of f (0) is
Let \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}, x \neq \frac{\pi}{4} .\] The value which should be assigned to f (x) at \[x = \frac{\pi}{4},\]so that it is continuous everywhere is
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos 10x}{x^2} , & x < 0 \\ a , & x = 0 \\ \frac{\sqrt{x}}{\sqrt{625 + \sqrt{x}} - 25}, & x > 0\end{cases}\] then the value of a so that f (x) may be continuous at x = 0, is
Find the values of a and b so that the function
If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is
If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
Let f (x) = |cos x|. Then,
The function f (x) = 1 + |cos x| is
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.
The point(s), at which the function f given by f(x) = `{("x"/|"x"|"," "x" < 0),(-1"," "x" ≥ 0):}` is continuous, is/are:
A real value of x satisfies `((3 - 4ix)/(3 + 4ix))` = α – iβ (α, β ∈ R), if α2 + β2 is equal to
The function f(x) = 5x – 3 is continuous at x =
For what value of `k` the following function is continuous at the indicated point
`f(x) = {{:(kx^2",", if x ≤ 2),(3",", if x > 2):}` at x = 2
Find the values of `a` and ` b` such that the function by:
`f(x) = {{:(5",", if x ≤ 2),(ax + b",", if 2 < x < 10),(21",", if x ≥ 10):}`
is a continuous function.
The function f(x) = x |x| is ______.
