MCQs of Limit and continuity questions (1)

1. For what value of $k$ the given function $\begin{cases} kx^2& \text{ if } x <4 \\ 4& \text{ if } x\geq4\end{cases}$ is continuous at $x=4$?

$\displaystyle \frac{1}{16}\\$
$\displaystyle \frac{1}{4}\\$
$\displaystyle 4\\$
$16\\$

2. If $\displaystyle f(x)= \begin{cases} ax+1& \text{ if } x \leq 3\\ bx+3& \text{ if } x>3\end{cases}$ is continuous then what is relation between $a$ and $b$ ?

$a+b=0\\$
$a+ \displaystyle \frac{1}{b}=3\\$
$a-\displaystyle \frac{1}{b}=3\\$
$a-b=\displaystyle \frac{2}{3}\\$

3. What is the value of $\lambda$ given function $f(x)=\displaystyle \begin{cases}\lambda(x^2-2x)& \text{ if } x \leq 0\\ 4x+1& \text{ if } x>0\end{cases}$ is continuous at $x=0$ ?

$\lambda = 2\\$
$\lambda = 0\\$
$\lambda = 1\\$
None of above

4. What is the value of $k$ the given function $\ f(x) =\displaystyle \begin{cases} \displaystyle \frac{k \cos x}{\pi -2x}& \text{ if } x \neq \displaystyle\frac{\pi}{2}\\ 3& \text{ if } x=\displaystyle \frac{\pi}{2}\end{cases}$ is continuous at $x=\displaystyle \frac{\pi}{2}$?

$k=2\\$
$k=-1\\$
$k=6\\$
None of above

5. For what values of $a$ and $b$ the function $\ f(x)=\displaystyle \begin{cases} 5& \text{ if } x \leq 2 \\ ax+b& \text{ if } 2 < x < 10 \\ 21& \text{ if } x \geq 10 \end{cases}$ is continuous ?

$a=2, b=1\\$
$a=2, b=2\\$
$a=1, b=2\\$
None of above

6. What is the value of $m$ the given function $\ f(x)=\begin{cases}mx+1 & \text{ if } x \leq 5 \\3x-5& \text{ if } x>5\end{cases}$ is continuous at $x=5$?

$\displaystyle \frac{5}{3}\\$
$\displaystyle \frac{9}{5}\\$
$\displaystyle \frac{3}{5}\\$
$\displaystyle \frac{5}{9}\\$

7. Find value of $k$ if function $f(x)=\begin{cases}\displaystyle\frac{\sin x}{x} & \text{ if } x \neq 0 \\k-1&\text{ if } x=0\end{cases}$ is continuous at $x=0$?

$2$
$0$
$-1$
$6$

8. Find value of $m$ if function $f(x) = \begin{cases}\displaystyle\frac{x^2-9}{x-9} & \text{ if } x \neq 3 \\m&\text{ if } x=3\end{cases}$ if it is continuous at $x=3$?

$3$
$6$
$2$
$1$

9. Find value of $m$ if function $f(x)=\displaystyle \begin{cases}\displaystyle\frac{x^3-27}{x-3} & \text{ if } x \neq 3 \\m&\text{ if } x=3\end{cases}$ is continuous at $x=3$ ?

$0$
$3$
$9$
$27$

10. What is the value of $m$ the function $f(x)=\displaystyle \begin{cases}\displaystyle\frac{x^3-8}{x-2} & \text{ if } x \neq 2 \\m&\text{ if } x=2\end{cases}$ is continuous at $x=2$ ?

$0$
$3$
$4$
$12$