Prove that for a-1-2- the function fx- asin- x-1 2 - x- 0 tan x -sin x x 3 - x-0 is continuous at x-0-

lim _ x→ 0 ^ f( x ) #160; =lim _ x→ 0 ^ asinπ( x+1 ) #160; 2 #160; #160; =asinπ #160; 2 #160; =a lim _ x→ 0 ^ + f( x ) ...

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Standard XII

Mathematics

Continuity of a Function

Prove that fo...

Question

Prove that for $a = 1 / 2$ , the function $f (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} a sin \frac{π (x + 1)}{2}; x \leq 0 \frac{tan x - sin x}{x^{3}}; x > 0 \end{matrix}$ is continuous at $x = 0$ .

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Solution

$lim x \to 0^{-} f (x) = lim x \to 0^{-} a sin \frac{π (x + 1)}{2}$
$= a sin \frac{π}{2} = a$
$lim x \to 0^{+} f (x) = lim x \to 0^{+} \frac{tan x - sin x}{x^{3}} \frac{0}{0}$ form
L-Hospital Rule
$= lim x \to 0^{+} \frac{{sec}^{2} x - cos x}{3 x^{2}} \frac{0}{0}$ form
$= lim x \to 0^{+} \frac{2 sec x (sec x tan x) + sin x}{6 x}$
$= lim x \to 0^{+} (\frac{{sec}^{2} x}{3} \times \frac{tan x}{x} + \frac{1}{6} \times \frac{sin x}{x})$
$= \frac{1}{3} + \frac{1}{6} = \frac{1}{2}$

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Similar questions

Q. Find

k

f (x) = {\begin{matrix} k sin \frac{π}{2}, & x \leq 0 \frac{tan x - sin x}{x^{3}}, & x > 0 \end{matrix}

is continuous at

x = 0

Q. Find the value of 'a' for which the function f defined as

f (x) = ⎧ ⎨ ⎩ \begin{matrix} a sin \frac{π}{2} (x + 1), x \leq 0 \frac{tan x - sin x}{x^{3}}, x > 0 \end{matrix} ⎫ ⎬ ⎭

is continuous at

x = 0

Q. Find the value of

^{'} a^{'}

for which

f (x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} k sin \frac{π}{2} (x + 1), x \leq 0 \frac{tan x - sin x}{x^{3}}, x > 0 \end{matrix}

is continuous at

x = 0

Q. Find the value of a if function f is defined as -

{a s i n \frac{π}{2} (x + 1), x \leq 0} \frac{t a n x - s i n x}{x^{3}}, x > 0

is continuous at x=0

Find the value of
$a$ for which the function f defined by

$f (x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} a s i n \frac{π}{2} (x + 1) & , x \leq 0 \frac{t a n x - s i n x}{x^{3}} & , x > 0 \end{matrix}$ is
continuous at $x = 0$

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Prove that for a=1/2, the function f(x)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩asinπ(x+1)2;x≤0tanx−sinxx3;x>0 is continuous at x=0.

limx→0−f(x)=limx→0−asinπ(x+1)2=asinπ2=alimx→0+f(x)=limx→0+tanx−sinxx300form L-Hospital Rule=limx→0+sec2x−cosx3x200form =limx→0+2secx(secxtanx)+sinx6x=limx→0+(sec2x3×tanxx+16×sinxx)=13+16=12

Prove that for $a = 1 / 2$ , the function $f (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} a sin \frac{π (x + 1)}{2}; x \leq 0 \frac{tan x - sin x}{x^{3}}; x > 0 \end{matrix}$ is continuous at $x = 0$ .