Maharashtra Board Solutions for Chapter: Continuity, Exercise 2: MISCELLANEOUS EXERCISE

Discuss the continuity of the following functions at the point(s) or on the interval indicated against them.

$f\left(x\right)=\left[x+1\right]$, for $x\in [-2, 2)$

Where $\left[.\right]$ is greatest integer function.

Discuss the continuity of the following function at the point(s) or on the interval indicated against them.

$f\left(x\right)=2x^2+x+1$, for $\left|x-3\right|\ge 2$

       $=x^2+3$,      for $1<x<5$.

Identify discontinuities if any for the following function as either a jump or a removable discontinuity on their respective domains.

$f\left(x\right)=x^2+x-3$, for $x\in [-5,-2)$

        $=x^2-5$,    for $x\in (-2,5]$.

Identify discontinuities if any for the following function as either a jump or a removable discontinuity on their respective domains.

$f\left(x\right)=x^2+5x+1$, for $0\le x\le 3$

$=x^3+x+5$, for $3<x\le 6$

Find $k$ if following function is continuous at the point indicated against them.

$f\left(x\right)={\left(\frac{5x-8}{8-3x}\right)}^{\frac{3}{2x-4}},$ for $x\ne 2$

          $=k$,  for $x=2$at $x=2$.

Find $k$ if following function is continuous at the point indicated against them.

$f\left(x\right)=\frac{{45}^x-9^x-5^x+1}{\left(k^x-1\right)\left(3^x-1\right)}$,   for $x\ne 0$

$=\frac{2}{3},$      for$x=0,$at $x=0$.

Find $f\left(a\right),$ if $f$ is continuous at $x=a$ where

$f\left(x\right)=\frac{1+\cos \left(\pi x\right)}{\pi (1-x{)}^2},$ for $x\ne 1$ and at$a=1.$

Find $f\left(\mathrm{\pi }\right),$ if $f$ is continuous at $x=\mathrm{\pi }$, where $f\left(x\right)=\frac{1-\cos \left[7\right(x-\pi \left)\right]}{5(x-\pi {)}^2},$ for $x\ne \pi$. (Write answer in fraction form)