Questions: Use the graph of the function (f) to determine the following limits, if they exist. (If an answer does not exist, enter DNE.). (lim x rightarrow infty f(x)=) (lim x rightarrow-infty f(x)=)

Use the graph of the function (f) to determine the following limits, if they exist. (If an answer does not exist, enter DNE.).

(lim x rightarrow infty f(x)=)

(lim x rightarrow-infty f(x)=)

Transcript text: Use the graph of the function $f$ to determine the following limits, if they exist. (If an answer does not exist, enter DNE.). \[ \lim _{x \rightarrow \infty} f(x)= \] $\square$ \[ \lim _{x \rightarrow-\infty} f(x)= \] $\square$

Solution

Solution Steps

Step 1: Identify the function and the limits to be evaluated

The function given is $ f(x) = \frac{x^2}{x^2 + 1} $. We need to determine the following limits:

$\lim_{x \to \infty} f(x)$
$\lim_{x \to -\infty} f(x)$

Step 2: Analyze the behavior of the function as $ x \to \infty $

As $ x \to \infty $, the term $ x^2 $ in both the numerator and the denominator dominates. Therefore, the function $ f(x) $ can be approximated by: \[ f(x) \approx \frac{x^2}{x^2} = 1 \]

Step 3: Evaluate the limit as $ x \to \infty $

Since $ f(x) $ approaches 1 as $ x \to \infty $: \[ \lim_{x \to \infty} f(x) = 1 \]

Step 4: Analyze the behavior of the function as $ x \to -\infty $

Similarly, as $ x \to -\infty $, the term $ x^2 $ in both the numerator and the denominator dominates. Therefore, the function $ f(x) $ can be approximated by: \[ f(x) \approx \frac{x^2}{x^2} = 1 \]

Step 5: Evaluate the limit as $ x \to -\infty $

Since $ f(x) $ approaches 1 as $ x \to -\infty $: \[ \lim_{x \to -\infty} f(x) = 1 \]