Theorems for Continuity

Q. (i) Evaluate ${lim}_{x\to \frac{\pi }{6}}\frac{\sqrt{3}sinx-cosx}{x-\frac{\pi }{6}}$.

(ii) If $f\left(x\right)=1+x+\frac{x^2}{2}+...+\frac{x^{100}}{100},$ then find the value of f'(1).

Q. (i) Evaluate ${lim}_{x\to 0}\frac{{10}^x-2^x-5^x+1}{xtanx}$.
(ii) Differentiate $\frac{1+tanx}{1-tanx}$ with respect to x.

Q.

Find the range of each of the following functions:

(i) $f\left(x\right)=2-3x,xϵR$ and $x>0$

(ii) $g\left(x\right)=x^2+2,xϵR$

Q.

Evaluate $\underset{x\to 0}{lim}\frac{(1+x{)}^6-1}{(1+x{)}^5-1}$

Q.

Evaluate $\underset{x\to 2}{lim}\left(\frac{x^5-32}{x^3-8}\right)$

Q.

If for non-zero x, $af\left(x\right)+bf\left(\frac{1}{x}\right)=\frac{1}{x}-5$, where $a\ne b$, find f(x).

Q.

What is a $T$ in linear algebra?

Q.

Find the range of each of the following functions:

(i) $f\left(x\right)=2-3x,xϵR,x>0$

(ii) $f\left(x\right)=x^2+2,x$ is a real number

(iii) $f\left(x\right)=x,x$ is a real number.

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

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

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. Let the function $f:R\to R$ be defined by $f\left(x\right)=x^3-x^2+(x-1)\sin x$ and let $g:R\to R$ be an arbitrary function. Let $fg:R\to R$ be the product function defined by $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right).$Then which of the following statements is/are TRUE?