#### To determine

**To show:**

The continuity of the given function F at x=−1.

#### Answer

F is continuous at x=−1.

#### Explanation

**Given:**

F(x)=bx+1−ax+1x+1

**Formulae used:**

For a given problem limx→cf(x)g(x), if the limits of f(x) and g(x) are 00 or ±∞∞ and if the limit of their derivatives limx→cf'(x)g'(x) is an extended real umber, then The L’ Hospital rule limx→0f(x)g(x)=limx→0f′(x)g′(x).

Consider F(x)=bx+1−ax+1x+1

For continuity limx→−1−F(x)=limx→−1+F(x)=F(−1)

Hence, use the formulae of the l’Hospital rule limx→0f(x)g(x)=limx→0f′(x)g′(x)

Now according to the given expression, the value of the left-hand limit is

limx→−1−bx+1−ax+1x+1=limx→−1−bx+1lnb−ax+1lna1=limx→−1−bx+1lnb−ax+1lna1=lnb−lna

And right hand limit is

limx→−1+bx+1−ax+1x+1=limx→−1+bx+1lnb−ax+1lna1=limx→−1+bx+1lnb−ax+1lna1=lnb−lna

F(−1)=lnb−lna

Thus F is continuous at x = -1