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 $\lim_{x\to -2^-}f(x)=3$.