Question

Given that f (x )is continuous. f (3 )equals 1, f (1 )equals 6, f (6 )equals negative 2, and f (negative 2 )equals 3. Determine limit as x rightwards arrow 6 to the power of plus of f (x ), and limit as x rightwards arrow negative 2 to the power of minus of f (x ).

Answer 1

Given:

`f(x)` is continuous,
`f(3)=1,f(1)=6,f(6)=-2,f(-2)=3`.

To find:

The value of
`\lim_(x\to 6^+)f(x)` and
`\lim_(x\to -2^-)f(x)`.

Solution:

If a function f(x) is continuous at
`x=c`, then

`\lim_(x\to c^-)f(x)=f(c)=\lim_(x\to c^+)f(x)`

It is given that the function
`f(x)` is continuous. It means it is continuous for each value and the left-hand and right-hand limits are equal to the value of the function.

The function is continuous for 6. So,

`\lim_(x\to 6^-)f(x)=f(6)=\lim_(x\to 6^+)f(x)`

`\lim_(x\to 6^+)f(x)=f(6)`

`\lim_(x\to 6^+)f(x)=-2`

The function is continuous for -2. So,

`\lim_(x\to -2^-)f(x)=f(-2)=\lim_(x\to -2^+)f(x)`

`\lim_(x\to -2^-)f(x)=f(-2)`

`\lim_(x\to -2^-)f(x)=3`

Therefore,
`\lim_(x\to 6^+)f(x)=-2` and
`\lim_(x\to -2^-)f(x)=3`.