Question

Help with these questions please!!

Answer 1

(1)

option-B

(2)

f(x) is continuous at a=4

Explanation:

(1)

we are given

`\lim_(x \to 0) (sin(2x))/(x)`

Since, we are suppose to find limit x-->0

so, we always choose value of x that is close to 0

At x=-0.03:

`(sin(2* (-0.03)))/((-0.03))=1.99880`

At x=-0.02:

`(sin(2* (-0.02)))/((-0.02))=1.99947`

At x=-0.01:

`(sin(2* (-0.01)))/((-0.01))=1.99987`

At x=0.01:

`(sin(2* (0.01)))/((0.01))=1.99987`

At x=0.02:

`(sin(2* (0.02)))/((0.02))=1.99947`

At x=0.03:

`(sin(2* (0.03)))/((0.03))=1.99880`

(2)

we are given

`f(x)=(x-4)/(x+5)`

Since, we have to check continuity at a=4

So, firstly we will find limit value and then functional value

Limit value:

`\lim_(x \to a)  f(x)=\lim_(x \to a)(x-4)/(x+5)`

now, we can plug a=4

`\lim_(x \to 4)  f(x)=\lim_(x \to 4)(4-4)/(4+5)`

`\lim_(x \to 4)  f(x)=0`

Functional value:

We can plug x=4 into f(x)

`f(4)=(4-4)/(4+5)`

`f(4)=0`

So, we can see that

`\lim_(x \to 4)  f(x)=f(4)=0`

So, limit value is equal to function value

so, f(x) is continuous at a=4.............Answer