Question

#Quality Question

# Continuity and Diffrentiability

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Answer 1

Step-by-step explanation:

Continuity and Differentiability is one of the most important topics which help students to understand the concepts like, continuity at a point, continuity on an interval, derivative of functions and many more. ... It implies that this function is not continuous at x=0

Answer 2

Question:

f(x) = {sin(a + 2)x + sinx}/x  ; x < 0

    = b    ; x = 0

    = ((x + 3x^2)^(1/3) - x^(1/3))/x^(4/3) ;x>0

is continuous at x = 0,  then a + 2b is equal to  (-2)

Answer:

$\sf{ \underline{For \:x <0: }}$

$\sf{\lim _{x \rightarrow0 {}^{ - } } \dfrac{sin(a + 2)x + sinx}{x}} \\\\ \sf {\lim _{x \rightarrow0 {}^{ - } } \dfrac{sin(a + 2)x}{x} + \lim _{x \rightarrow0 {}^{ - } }\dfrac{sinx}{x}} \\\\ \sf{ \lim _{x \rightarrow0 {}^{ - } }(a + 2) \dfrac{sin(a + 2)x}{(a + 2)x} + \lim _{x \rightarrow0 {}^{ - } }\dfrac{sinx}{x}} \\\\ \sf{(a + 2)\lim _{x \rightarrow0 {}^{ - } }\dfrac{sin(a + 2)x}{(a + 2)x} + \lim _{x \rightarrow0 {}^{ - } }\dfrac{sinx}{x} } \\\\(a + 2)(1) + 1 \\ a + 3$

$\sf{\underline{For \: x = 0:} f(x) = b}$

$\sf{ \underline{For \: x > 0 : }}$

$\lim _{x \rightarrow0 {}^{ + } } \frac{(x + 3x {}^{2}) {}^{ \frac{1}{3} } - x {}^{ \frac{1}{3} }}{x {}^{ \frac{4}{3}} } \\\\ \lim _{x \rightarrow0 {}^{ + } } \frac{x {}^{ \frac{1}{3} } (1 + 3x {}^{}) {}^{ \frac{1}{3} } - x {}^{ \frac{1}{3}}}{x {}^{ \frac{4}{3}} } \\\\ \lim _{x \rightarrow0 {}^{ + } } \frac{x {}^{ \frac{1}{3} } ((1 + 3x {}^{}) {}^{ \frac{1}{3} } - 1 ){}^{ \frac{}{}}}{x {}^{ \frac{4}{3}} } \\\\ \lim _{x \rightarrow0 {}^{ + } } \frac{(1 + 3x) {}^{ \frac{1}{3} } - 1}{x}$

This is an intermediate form. Using L-Höpital rule:

$\lim _{x \rightarrow0 {}^{ + } } \frac{ \frac{1}{3} (1 + 3x) {}^{ \frac{1}{3} -1} + 0}{1} = \frac{1}{3}$

$\sf{Therefore, for \: continuity \: at \: x = 0, } \\ \lim _{x \rightarrow0 {}^{ - } }f(0) = \lim _{x \rightarrow0 {}^{ + } }f(0) = f(0) \\ a + 3 = b = \frac{1}{3}$

From here we get, a = - 8/3 and b = 1/3

Hence, a + 2b = - 8/3 + 2(1/3)

                        = - 2