Examine the continuity of the function f (x) = x²+5 at x = -1

1

(a)

f( -1 ) = 1 + 5 = 6,
\(\lim _{x \rightarrow-1} f(x)=\lim _{x \rightarrow-1}\left(x^{2}+5\right)=1+5=6\)
\(\text { As } \lim _{x \rightarrow-1} f(x)=f(-1)\)
Hence, continuous at x =-1.

Examine the continuity of the function f (x) = \(\frac { 1 }{ x+3 } , x\ \in \ R\).

1

(a)

For x=-3 function is not defined. Hence, not continuous for \(x\ \in \ R\).

Give an example of a function which is continuous at x = 1, but not differentiable at x = 1.

1

(a)

Absolute value function fix) = Ix - 1I is continuous at x = 1 but not differentiable at x = 1.

State the points of discountinuity for the function \(f(x)= [x]\)  in \(-3 < x < 3.\)

1

(a)

f(x) = [x]  is not continuous for integers.Hence not continuous at \(x=\pm 2, \pm 1,0\)

Discuss the continuity of the function defined by f(x) = \(\frac { 1 }{ x } ,\quad x\neq 0\)

1

(a)

Continuous at x=a (\(\neq \)0) \(\in \) R.

Check the continuity of the function f(x) = {\(\begin{matrix} 1, & if & x\le 0 \\ 2, & if & x>0 \end{matrix}\)

1

(a)

{Not continuous}

Check the continuity of the function f (x) = |x| at x = 0

1

(a)

{continuous}

Check the continuity of the function f(x) = sinx + x² at x = 0

1

(a)

{continuous}

Examine the continuity of the function f(x) = |x|

1

(a)

{continuous}

Write the points of discontinuity of the function f(x) = [x] where [x] denotes the gretest integer function less than or equal to x.

1

(a)

{All Integers}