# How do you find a value k such that the limit exists given #(x^2+8x+k)/(x+2)#?

##### 1 Answer

If we define:

If

For any other value of alpha, the limit always exist,

# L(alpha) = (alpha^2+8alpha+k)/(alpha+2) #

#### Explanation:

The question is incomplete as the value of

Let us consider the function:

# L(alpha) = lim_(x rarr alpha) (x^2+8x+k)/(x+2) #

The limit function is not defined when the denominator is zero, ie when

So let us consider the case

# L(-2) = lim_(x rarr -2) (x^2+8x+k)/(x+2) #

This limit will exist iff the numerator also has a factor of

# (x+beta)(x+2) -= x^2+8x+k #

# =>x^2+(2+beta)x+2beta = x^2+8x+k #

If we equate coefficients then we get:

# x^1 : 2+beta = 8 => beta = 6 #

# x^0 : 2beta = k => k=12 #

Hence, we have a factor that will cancel with

# L(-2) = lim_(x rarr -2) (x^2+8x+12)/(x+2) #

# " " = lim_(x rarr -2) ((x+6)(x+2))/(x+2) #

# " " = lim_(x rarr -2) (x+6) #

# " " = 4 #

For any other value of

# L(alpha) = lim_(x rarr alpha) (x^2+8x+k)/(x+2) #

# " " = (alpha^2+8alpha+k)/(alpha+2) #