Continuity at x=0

To determine if the function f(x) is continuous at x=0, we need to check three conditions:
1. The function f(x) is defined at x=0.
2. The limit of f(x) as x approaches 0 exists.
3. The value of f(x) at x=0 is equal to the limit.

Condition 1: The function is defined at x=0

Since f(x) is defined as K when x=0, the function is defined at x=0.

Condition 2: The limit of f(x) as x approaches 0 exists

To find the limit of f(x) as x approaches 0, we can use L'Hopital's Rule.

Taking the limit of f(x) as x approaches 0:
lim (x→0) [sin^2(x) - tan^2(x)] / [(e^x - 1) * x^2]

Applying L'Hopital's Rule, we differentiate the numerator and denominator with respect to x:

lim (x→0) [2sin(x)cos(x) - 2tan(x)sec^2(x)] / [(e^x - 1) * 2x]

Since the limit is still indeterminate, we can apply L'Hopital's Rule again:

lim (x→0) [2cos^2(x) - 2sec^4(x) + 2tan^2(x)sec^2(x)] / [(e^x - 1) * 2]

Now, we can substitute x=0 into the expression:

[2cos^2(0) - 2sec^4(0) + 2tan^2(0)sec^2(0)] / [(e^0 - 1) * 2]

Simplifying further, we get:

[2 - 2(1) + 0] / [1 * 2]
= 0

Therefore, the limit of f(x) as x approaches 0 exists and is equal to 0.

Condition 3: The value of f(x) at x=0 is equal to the limit

Since f(x) is defined as K when x=0, we have:

f(0) = K

And the limit of f(x) as x approaches 0 is 0. So, we have:

K = 0

Therefore, the absolute value of K is |0| = 0.

Conclusion:

The function f(x) is continuous at x=0 and the absolute value of K is 0.

Hey Aarohi'
is the answer k=-2