If f x = sin 3 x + A sin 2 x + B sin x x 5 x ≠ 0 is continuous at x = 0 . Find A & B . Also, find f 0 .

Given, f x = sin 3 x + A sin 2 x + B sin x x 5 x ≠ 0 is continuous at x = 0 . That means lim x → 0 f x exist, lim x → 0 sin 3 x + A sin 2 x + B sin x x 5 0 0 an indeterminate form, then using L'Hospital rule, ⇒ lim x → 0 3 cos 3 x + 2 A cos 2 x + B cos x 5 x 4 for existing limit, the numerator should be zero at x → 0 to make 0 0 an indeterminate form. We get, 3 + 2 A + B = 0 . . . i Again using L'Hospital rule, then we get, lim x → 0 - 9 sin 3 x - 4 A sin 2 x - B sin x 20 x 3 0 0 an indeterminate form use L'Hospital rule, ⇒ lim x → 0 - 27 cos 3 x - 8 A cos 2 x - B cos x 60 x 2 again for the existing limit numerator should be zero x → 0 to make an indeterminate form. We get, - 27 - 8 A - B = 0 . . . ii Now solving i & ii , we get, A = - 4 , B = 5 . Again using L'Hospital rule, lim x → 0 81 sin 3 x - 64 sin 2 x + 5 sin x 120 x 0 0 an indeterminate form again using L'Hospital rule, then we get, lim x → 0 243 cos 3 x - 128 cos 2 x + 5 cos x 120 = 1

If the graph of the function y = f ( x ) is and the graph of the function y = g ( x ) is Then discuss the continuity of f x g x at x = 3 and x = 2 .

Now given the graph of y = f x and the graph of y = g x Check continuity of f x · g x at x = 2 , x = 3 . LHL = RHL = f x · g x i Check at x = 2 LHL = f x · g x = 2 · 1 = 2 (from the graph) and RHL = f x · g x = 1 · 2 = 2 (from the graph) f 2 · g 2 = 4 (from the graph) LHL = RHL ≠ f 2 · g 2 , then f x · g x is discontinuous at x = 2 . ii Check at x = 3 LHL = f x · g x = 2 (from the graph) f 3 · g 3 = 1 . 2 = 2 (from the graph) RHL data is not present in the graph on the basis of LHL = f 3 · g 3 = 2 Then, f x · g x is continuous at x = 3 .

Let f : [ a , b ] → [ 1 , ∞ ) be a continuous function and let g : R → R be defined as g x = 0 i f x < 0 ∫ a x f t d t i f a ≤ x ≤ b ∫ a b f t d t i f x > b . Then :

g x is continuous but not differentiable at b

g x is continuous and differentiable at either a or b but not both

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 20

EASY

JEE Main/Advance

IMPORTANT

Earn 100

If $f (x) = \{x\} & g (x) = [x]$ (where $\{.\} & [.]$ denotes the fractional part and the integral part functions respectively), then discuss the continuity of $h (x) = f (x) + g (x)$ at $x = 1 .$

Important Questions on Continuity and Differentiability

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 21

If $f (x) = \{x\} & g (x) = [x]$ (where $\{.\} & [.]$ denotes the fractional part and the integral part functions respectively), then discuss the continuity of $h (x) = f (x) - g (x)$ at $x = 1 .$

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 22

If $f (x) = \{x\} & g (x) = [x]$ (where $\{.\} & [.]$ denotes the fractional part and the integral part functions respectively), then discuss the continuity of $h (x) = g (x) + \sqrt{f (x)}$ at $x = 1$ and $x = 2 .$

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 26 - Continuity and Differentiability>Exercise-1>Q 23

Suppose that $f (x) = x^{3} - 3 x^{2} - 4 x + 12$ and $h (x) = \{\begin{cases} \begin{matrix} \frac{f (x)}{x - 3} & , x \neq 3 \end{matrix} \\ K, x = 3 \end{cases}$ , then

$(a)$ Find all zeros of $f (x) .$
$(b)$ Find the value of $K$ that makes $h (x)$ continuous at $x = 3 .$
$(c)$ Using the value of $K$ found in $(b),$ determine whether $h (x)$ is an even function.