Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).
f(x) = x²(4x² − √(16x⁴ + 1))

Verified step by step guidance

1. Step 1: Identify the points where the function might have vertical asymptotes. These occur where the function is undefined or where the denominator is zero. In this case, since there is no denominator, we need to consider where the expression inside the square root becomes zero or negative.
2. Step 2: Analyze the expression inside the square root: \(16x^4 + 1\). Since this expression is always positive for all real \(x\), there are no points where the function is undefined due to the square root.
3. Step 3: Since the function \(f(x) = x^2(4x^2 - \sqrt{16x^4 + 1})\) does not have a denominator that can become zero, we need to consider the behavior of the function as \(x\) approaches infinity or negative infinity to check for vertical asymptotes.
4. Step 4: Evaluate the limits \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\) for any potential vertical asymptotes. Since there are no points where the function is undefined, we focus on the behavior at infinity.
5. Step 5: Conclude that since there are no values of \(x\) that make the function undefined or cause a division by zero, there are no vertical asymptotes for this function.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value from either the left or the right. This typically happens at points where the function is undefined, often due to division by zero. Identifying these points is crucial for understanding the behavior of the function near those values.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, evaluating the left-hand limit (lim x→a^- f(x)) and the right-hand limit (lim x→a⁺ f(x)) helps determine the behavior of the function near the asymptote, indicating whether it approaches positive or negative infinity.

Function Behavior

Understanding the behavior of a function involves analyzing how it changes as the input varies, particularly near critical points like vertical asymptotes. This includes observing trends such as increasing or decreasing values, and how the function behaves as it approaches the asymptote from either side, which is essential for sketching the graph and predicting the function's overall shape.