Question

If f(x) = `{{:((x^2 - 4x - 5)/(x + 1)",", x ≠ -1), (k",",x = -1):}`

is continuous at x = −1, then the value of k is:

विकल्प

• Any real value

• 6

• −1

• −6

Answer 1

−6

Explanation:

f(x) to be continuous at x = −1

`lim_(x->-1) f(x) = f(-1)`

We know that f(−1) = k

So, we need to calculate the limit:

`lim_(x->-1) (x^2 - 4x - 5)/(x + 1) = k`

Factor the numerator:

x² − 4x − 5

Need two numbers that multiply to −5 and add to −4. These are −5 and +1

So,

x² − 4x − 5 = (x − 5)(x + 1)

Substitute this back into the limit:

`lim_(x->-1) ((x - 5)(x + 1))/(x + 1)`

Since x approaches −1 but is not equal to it, we can cancel the (x + 1) terms:

`lim_(x->-1) (x - 5) = -1 - 5 = -6`

For the function to be continuous, the value of the function at x = −1 must equal this limit:

k = −6