Theorems for Continuity

Q. If h(x) = g(x) + f(x) is a continuous function in a given interval then g(x) and f(x) individually will also be continuous in the same interval.

1. True

2. False

Q.

If the function 'f' and 'g' are continuous at c then. Choose the incorrect alternative.

1. f + g is continuous at c

2. f - g is continuous at c

3. is continuous at c

4. f.g is continuous at c

Q.

Given $f\left(x\right)=g\left(x\right).h\left(x\right)$. If $g\left(x\right)$ and $h\left(x\right)$ are continuous in a given interval then $f\left(x\right)$ would also be continuous in the same interval.

1. True

2. False

Q.

f(x) and g(x) are continuous functions such that ${lim}_{x\to a}\left[3f\right(x)+g(x\left)\right]=6and{lim}_{x\to a}\left[2f\right(x)-g(x\left)\right]=4$. Given that the function h(x).g(x) is continuous at x = a and h(a)=4, which of the following must be true?

1. ${lim}_{x\to a^{-}}h\left(x\right)ϵR$ and ${lim}_{x\to a^{+}}h\left(x\right)ϵR$

2. ${lim}_{x\to a}h\left(x\right)=0$

3. ${lim}_{x\to a}h\left(x\right)=4$

4. ${lim}_{x\to a}h\left(x\right)=\infty$

Q.

f(x) and g(x) are continuous functions such that ${lim}_{x\to a}\left[3f\right(x)+g(x\left)\right]=6and{lim}_{x\to a}\left[2f\right(x)-g(x\left)\right]=4$. Given that the function h(x).g(x) is continuous at x = a and h(a)=4, which of the following must be true?

1. ${lim}_{x\to a}h\left(x\right)=\infty$

2. ${lim}_{x\to a^{-}}h\left(x\right)ϵR$ and ${lim}_{x\to a^{+}}h\left(x\right)ϵR$

3. ${lim}_{x\to a}h\left(x\right)=4$

4. ${lim}_{x\to a}h\left(x\right)=0$

Q. x+1, if x2110, f(x)=

Q. If $f\left(x\right)=\{\begin{array}{c}\frac{\sin \left(\cos x\right)-\cos x}{{(\pi -2x)}^3},if\\ k\end{array}\begin{array}{c}x\ne \frac{\pi }{2}\\ ,ifx=\frac{\pi }{2}\end{array}$ is continuous at $x=\frac{\pi }{2}$, then $k=$

1. $0$
2. $-\frac{1}{2}$
3. $-\frac{1}{24}$
4. $-\frac{1}{48}$

Q. $f\left(x\right)=\{\begin{array}{c}\frac{1-{\sin }^{2}x}{3{\cos }^{2}x},\\ a,\\ \frac{b(1-\sin x)}{{(\pi -2x)}^2}\end{array}\begin{array}{c}x<\frac{\pi }{2}\\ x=\frac{\pi }{2}\\ x>\frac{\pi }{2}\end{array}$,
then $f\left(x\right)$ is continuous at $x=\frac{\pi }{2}$, if

Q.
The function $f\left(x\right)=\frac{\cos x-\sin x}{\cos 2x}$ is not defined at $x=\frac{\pi }{4}$ The value of $f\left(\frac{\pi }{4}\right)$ so that $f\left(x\right)$ is continuous at $x=\frac{\pi }{4}$ is

1. $\frac{1}{\sqrt{2}}$
2. $\sqrt{2}$
3. $-\sqrt{2}$
4. $1$

Q. Thefunction $f\left(x\right)=x^2\sin \frac{1}{x},x\ne 0,f\left(0\right)=0$ at $x=0$

1. Is continuous but not differentiable
2. Is having continuous derivative
3. Is continuous
4. Is continuous and differentiable