Let f(x) = begin{cases} (1 + |sin x|)^{a/|sin | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) For $f(x)$ to be continuous at $x = 0$,we must have $\lim_{x 	o 0^-} f(x) = f(0) = \lim_{x 	o 0^+} f(x)$.First,calculate the left-hand limit $(L

Vedclass · Accepted Answer

$2/3, e^{2/3}$

Vedclass · Answer

(B) For $f(x)$ to be continuous at $x = 0$,we must have $\lim_{x 	o 0^-} f(x) = f(0) = \lim_{x 	o 0^+} f(x)$.First,calculate the left-hand limit $(LHL)$:$\lim_{x 	o 0^-} (1 + |\sin x|)^{a/|\sin x|} = e^{\lim_{x 	o 0^-} |\sin x| \cdot \frac{a}{|\sin x|}} = e^a$.Next,calculate the right-hand limit $(RHL)$:$\lim_{x 	o 0^+} e^{	an 2x/	an 3x} = e^{\lim_{x 	o 0^+} \frac{	an 2x}{	an 3x}} = e^{\lim_{x 	o 0^+} \frac{(	an 2x / 2x) \cdot 2x}{(	an 3x / 3x) \cdot 3x}} = e^{2/3}$.Since $f(0) = b$,we equate the limits:$e^a = b = e^{2/3}$.Thus,$a = 2/3$ and $b = e^{2/3}$.

Let $f(x) = \begin{cases} (1 + |\sin x|)^{a/|\sin x|}, & -\pi/6 < x < 0 \\ b, & x = 0 \\ e^{\tan 2x/\tan 3x}, & 0 < x < \pi/6 \end{cases}$. If $f$ is continuous at $x = 0$,then the values of $a$ and $b$ are respectively:

Similar Questions

If the function $f(x) = \begin{cases} \frac{x^2-(A+2)x+A}{x-2} & \text{for } x \neq 2 \\ 2 & \text{for } x=2 \end{cases}$ is continuous at $x=2$, then:

Let $S_n = 1 + 3x + 9x^2 + 27x^3 + \ldots$ ($n$ terms) and $-\frac{1}{3} < x < \frac{1}{3}$. If $\lim_{n \rightarrow \infty} S_n = f(x)$,then $f(x)$ is discontinuous at the point $x =$

For every pair of continuous functions $f, g: [0, 1] \rightarrow \mathbb{R}$ such that $\max \{f(x): x \in [0, 1] \} = \max \{g(x): x \in [0, 1] \} = \lambda$,the correct statement$(s)$ is (are):

If the function $f(x)$ defined by $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$,then $k = . . . . . .$

If $f: R \rightarrow R$ defined by $f(x) = \begin{cases} \frac{1 + 3 x^2 - \cos 2 x}{x^2}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$, then $k$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module