Question

Limits of a Function

If
`\displaystyle \lim_(x \to 1) (f(x)-8)/(x-1) = 10,` find
`\lim_(x \to 1) f(x).`

Answer 1

8.

Explanation:

lim x --> 1 of 10x - 10 / (x - 1)

= lim x -->1 of 10/1 ( Because when x = 1 the function = 0/0 so we can apply

L'hopitals rule which states that that if f(x)/ g(x) = 0/0 when x = limit value, then the limit is = f'(x) / g'(x))

So f(x) - 8 = 10x - 10

f(x) = 10x - 10 + 8

f(x) = 10x - 2

- and the limit as x ---> 1 of this is

10(1) - 2

= 8.

Answer 2

We are given the equation:

`\lim_(x \to 1) (f(x)-8)/(x-1) = 10`

which can be rewritten as:

`(\lim_(x \to 1)[f(x)-8])/(\lim_(x \to 1)[x-1]) = 10`

`\lim_(x \to 1)[f(x)-8] = 10*\lim_(x \to 1)[x-1]`

`\lim_(x \to 1)[f(x)-8] = \lim_(x \to 1)[10x-10]`

which means that..

`f(x) - 8 = 10x - 10`

`f(x) = 10x - 10 + 8`

`f(x) = 10x - 2`

Finding the limit:

`\lim_(x \to 1) f(x) = \lim_(x \to 1) (10x - 2)`

`\lim_(x \to 1) f(x) = 10(1) - 2`

`\lim_(x \to 1) f(x) = 8`