Questions: Find the limit of f(x) as x approaches 0 from the left, the limit of f(x) as x approaches 0 from the right, and the limit of f(x) as x approaches 0 numerically if they exist. Round each answer to the fourth decimal place. If a limit does not exist, enter DNE. f(x) = (10x) / (x^2 + 5x) x: -0.1, -0.01, -0.001, -0.0001, 0, 0.0001, 0.001, 0.01, 0.1 f(x):

Find the limit of f(x) as x approaches 0 from the left, the limit of f(x) as x approaches 0 from the right, and the limit of f(x) as x approaches 0 numerically if they exist.
Round each answer to the fourth decimal place. If a limit does not exist, enter DNE.

f(x) = (10x) / (x^2 + 5x)

x: -0.1, -0.01, -0.001, -0.0001, 0, 0.0001, 0.001, 0.01, 0.1
f(x):

Transcript text: Find $\lim _{x \rightarrow 0^{-}} f(x), \lim _{x \rightarrow 0^{+}} f(x)$ and $\lim _{x \rightarrow 0} f(x)$ numerically if they exist. Round each answer to the fourth decimal place. If a limit does not exist, enter DNE. \[ f(x)=\frac{10 x}{x^{2}+5 x} \] \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline$x$ & -0.1 & -0.01 & -0.001 & -0.0001 & 0 & 0.0001 & 0.001 & 0.01 & 0.1 \\ \hline$f(x)$ & & & & & - & & & & \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Evaluate $ f(x) $ as $ x $ Approaches 0 from the Left

We calculate $ f(x) $ for values of $ x $ approaching 0 from the left: \[ x = -0.1, -0.01, -0.001, -0.0001 \] The corresponding function values are: \[ f(-0.1) \approx 2.0408, \quad f(-0.01) \approx 2.004, \quad f(-0.001) \approx 2.0004, \quad f(-0.0001) \approx 2.0000 \]

Step 2: Evaluate $ f(x) $ as $ x $ Approaches 0 from the Right

Next, we calculate $ f(x) $ for values of $ x $ approaching 0 from the right: \[ x = 0.0001, 0.001, 0.01, 0.1 \] The corresponding function values are: \[ f(0.0001) \approx 2.0000, \quad f(0.001) \approx 1.9996, \quad f(0.01) \approx 1.9960, \quad f(0.1) \approx 1.9608 \]

Step 3: Determine the Left-Hand and Right-Hand Limits

From the calculations, we find: \[ \lim_{x \to 0^{-}} f(x) \approx 2.0000 \quad \text{and} \quad \lim_{x \to 0^{+}} f(x) \approx 1.9999 \] Since the left-hand limit $ \lim_{x \to 0^{-}} f(x) $ and the right-hand limit $ \lim_{x \to 0^{+}} f(x) $ are approximately equal, we conclude that: \[ \lim_{x \to 0} f(x) = 2.0 \]