Limit

Q. If $\underset{x\to 0}{lim}\varphi \left(x\right)=a^3,a\ne 0$, then $\underset{x\to 0}{lim}\varphi \left(\frac{x}{a}\right)$ is

1. $a^2$
2. $\frac{1}{a^3}$
3. $\frac{1}{a^2}$
4. $a^3$

Q. Let $f\left(x\right)=\frac{3}{4}x+1,$ and $f^n\left(x\right)$ be defined as $f^2\left(x\right)=f\left(f\right(x\left)\right)$ and for $n\ge 2,f^{n+1}\left(x\right)=f\left(f^n\right(x\left)\right).$ If $\lambda =\underset{n\to \infty }{lim}{f}^n\left(x\right),$ then

1. $\lambda$ is independent of $x$
2. $\lambda$ is a linear polynomial in $x$
3. the line $y=\lambda$ has slope $0$
4. the line $4y=\lambda$ touches the unit circle with centre at the origin

Q.

$li{m}_{x\to \frac{\pi }{4}}\frac{1-co{t}^{3}x}{2-cotx-co{t}^{3}x}=$___

1. 0

2. $\frac{3}{4}$

3. $\frac{1}{2}$

4. $\infty$

Q.

f(x) = [x], the greatest integer less then or equal to x and k is an integer. Then, ${lim}_{x\to k}f\left(x\right)=$___

1. k

2. k - 1

3. k + 1

4. does not exist