Problem 61 - Calculus: Early Transcendentals

Question

Find the limits as \( x \to \infty \) and as \( x \to -\infty \). Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. \( y = x^4 - x^6 \)

Step-by-Step Solution

Answer

As \( x \to \pm\infty \), \( y \to \infty \); x-intercepts are -1, 0, 1.

Step 1: Factor the Expression

Notice that the expression can be factored to simplify analysis. Factor out the highest power of \(x\) common to both terms: \( y = x^6 (1 - \frac{1}{x^2}) \). This makes it easier to evaluate the limits as \(x\) approaches infinity or negative infinity.

Step 2: Find the Limit as \( x \to \infty \)

Consider the factorized expression \( y = x^6 (1 - \frac{1}{x^2}) \). As \( x \to \infty \), \( \frac{1}{x^2} \to 0 \). Hence, \( y \approx x^6 \cdot 1 = x^6 \). Therefore, \( \lim_{x \to \infty} y = \infty \).

Step 3: Find the Limit as \( x \to - fty \)

Now, consider the factorized expression for \( x \to -\infty \). Again, as \( x \to -\infty \), \( \frac{1}{x^2} \to 0 \); therefore, \( y \approx (-x)^6 \). Since any even power of a negative number is positive, \( \lim_{x \to -\infty} y = \infty \).

Step 4: Determine x-intercepts and y-intercept

To find x-intercepts, set \( y = 0 \). Solve: \( x^4 - x^6 = 0 \) which factors to \( x^4(1 - x^2) = 0 \). Thus, \( x = 0 \) or \( x = \pm 1 \). For the y-intercept, set \( x = 0 \), giving \( y = 0^4 - 0^6 = 0 \).

Step 5: Sketch a Rough Graph

Combine all the information: The curve approaches \( y = \infty \) as \( x \to \infty \) and \( x \to -\infty \). It intersects the x-axis at \( x = -1, 0, 1 \) and the y-axis at \( y = 0 \). The behavior near these points shows the curve rises steeply at both ends.

Key Concepts

Infinite Limits

Infinite limits in calculus describe what happens to a function's value as the input either increases or decreases without bound. In this exercise, we deal with a polynomial function:

• The goal is to find the limit of the function as \( x \to \infty \) and \( x \to -\infty \).

To do this, one can factor the expression: \( y = x^6 (1 - \frac{1}{x^2}) \). As \( x \) becomes extremely large or extremely small, \( \frac{1}{x^2} \) nearly becomes zero. Consequently, the term \( y \) approximates \( x^6 \) as \( x \to \infty \) and \( y \approx (-x)^6 \) as \( x \to -\infty \), both leading to infinite values. This indicates that the function grows without bound in either direction of the x-axis.

Intercepts

Intercepts are crucial for understanding where a graph crosses the axes, providing insight into its behavior. There are two main types of intercepts:

• X-intercepts: These are the points where the graph crosses the x-axis, which occurs when \( y = 0 \).

By setting \( x^4 - x^6 = 0 \) and factoring, it becomes \( x^4(1 - x^2) = 0 \). Thus, the x-intercepts occur at \( x = 0 \), \( x = -1 \), and \( x = 1 \).

• Y-intercept: At this point the graph crosses the y-axis, which happens when \( x = 0 \).

Plugging \( x = 0 \) into the equation gives \( y = 0 \). Therefore, the y-intercept is at \( (0, 0) \).

Graph Sketching

Graph sketching is a method used to draw a graph based on important characteristics like limits, intercepts, and general behavior. Here's how to sketch based on our previous analysis:

• As \( x \to \infty \) and \( x \to -\infty \), the function tends to \( \infty \). This indicates that the graph rises steeply on both sides.

The x-intercepts and the y-intercept give anchor points where the graph touches the axes.

• At \( x = -1, 0, 1 \), the graph crosses the x-axis.
• From these points and the infinite behavior, the graph appears as a 'U' shape, rising on both ends, appearing symmetrical.

Adding this together forms a rough sketch that aligns with polynomial curve characteristics given by the function.

Even Functions

An even function is symmetric with respect to the y-axis. This symmetry means if you flip the graph over the y-axis, it remains unchanged.
In mathematical terms, a function \( f(x) \) is even if \( f(x) = f(-x) \) for all \( x \).

• For our function \( y = x^4 - x^6 \), check if \( f(-x) = (-x)^4 - (-x)^6 = x^4 - x^6 = f(x) \).

Therefore, the function is even.

• This confirms why the graph is symmetric around the y-axis, reaffirming our earlier findings from the sketch.
• The symmetry helps ensure that the sketch of this even function is balanced and accurate.