Problem 56 - Calculus Early Transcendentals

Question

$$ \begin{array}{l}{\text { Make a rough sketch of the curve } y=x^{n}(n \text { an integer })} \\ {\text { for the following five cases: }}\end{array} $$ $$ \begin{array}{ll}{\text { (i) } \mathrm{n}=0} & {\text { (ii) } \mathrm{n}>0, \mathrm{n} \text { odd }} \\ {\text { (iii) } \mathrm{n}>0, \mathrm{n} \text { even }} & {\text { (iv) } \mathrm{n}<0, \mathrm{n} \text { odd }}\end{array} $$ $$ \begin{array}{l}{\text { (v) } \mathrm{n}<0, \text { n even }} \\ {\text { Then use these sketches to find the following limits. }}\end{array} $$ (a) $$ \lim _{x \rightarrow \mathbb{u}^{+}} x^{n} \quad \text { (b) } \lim _{x \rightarrow 0^{-}} x^{n} $$ (c) $$ \lim _{x \rightarrow \infty} x^{n} \quad \text { (d) } \lim _{x \rightarrow-\infty} x^{n} $$

Step-by-Step Solution

Answer

The limits are: (a) 1 or 0 or ∞, (b) 1 or 0 or -∞ or ∞, (c) 1 or ∞ or 0, (d) 1 or -∞ or ∞ or 0.

Step 1: Sketching for n = 0

For \( n = 0 \), the function becomes \( y = x^0 = 1 \) for any \( x \). This is a horizontal line across the y-axis at \( y=1 \).

Step 2: Sketching for n > 0, n odd

For any positive odd \( n \), like \( n = 1, 3, 5, ... \), the graph of \( y = x^n \) is similar to \( y = x^3 \). It passes through the origin, is symmetric with respect to the origin, and extends to positive infinity for positive \( x \) and to negative infinity for negative \( x \).

Step 3: Sketching for n > 0, n even

For positive even \( n \), like \( n = 2, 4, 6, ... \), the graph of \( y = x^n \) is similar to \( y = x^2 \). It is symmetric about the y-axis, passing through the origin, and both branches go to positive infinity as \( |x| \) increases.

Step 4: Sketching for n < 0, n odd

For negative odd \( n \), like \( n = -1, -3, -5, ... \), the function resembles \( y = \frac{1}{x} \). As \( x \) approaches 0 from the positive side, the curve goes to positive infinity, and as \( x \) approaches 0 from the negative side, it goes to negative infinity.

Step 5: Sketching for n < 0, n even

For negative even \( n \), like \( n = -2, -4, -6, ... \), the graph is similar to \( y = \frac{1}{x^2} \). Approaching 0 from either direction, the curve goes to positive infinity, and as \( |x| \) increases, \( y \) approaches zero on both sides.

Step 6: Solving limits based on sketches - Part (a)

For \( \lim_{x \rightarrow 0^{+}} x^n \), - \( n = 0 \): Limit is 1. - \( n > 0, \text{ odd/even} \): Limit is 0. - \( n < 0, \text{ odd/even} \): Limit is \( \infty \).

Step 7: Solving limits based on sketches - Part (b)

For \( \lim_{x \rightarrow 0^{-}} x^n \), - \( n = 0 \): Limit is 1. - \( n > 0, \text{ odd} \): Limit is 0. - \( n > 0, \text{ even} \): Limit is 0. - \( n < 0, \text{ odd} \): Limit is \( -\infty \). - \( n < 0, \text{ even} \): Limit is \( \infty \).

Step 8: Solving limits based on sketches - Part (c)

For \( \lim_{x \rightarrow \infty} x^n \), - \( n = 0 \): Limit is 1. - \( n > 0 \): Limit is \( \infty \). - \( n < 0 \): Limit is 0.

Step 9: Solving limits based on sketches - Part (d)

For \( \lim_{x \rightarrow -\infty} x^n \), - \( n = 0 \): Limit is 1. - \( n > 0, \text{ odd} \): Limit is \( -\infty \). - \( n > 0, \text{ even} \): Limit is \( \infty \). - \( n < 0 \): Limit is 0.

Key Concepts

Limits

In mathematical terms, limits help us understand what happens to a function as the input approaches a certain value. Limits are particularly useful in understanding the behavior of polynomial functions as their input values become very large, very small, or approach zero. Let's break it down a bit more.

• For any polynomial, say \( x^n \), if \( n \) is positive, the limit as \( x \) approaches zero from positive or negative infinity is zero. This is because as \( x \) becomes vastly large, the value of \( x^n \) shoots up or descends drastically, depending on the sign of \( n \).


• If \( n \) is zero, \( x^0 \) equals 1 for any \( x \), and thus the limit is 1 in all those circumstances, because the function is constant.


• For negative \( n \), as \( x \) approaches zero, \( x^n \) can become very large or small, indicating an infinite limit depending on the direction \( x \) is approaching zero from.

The limit at infinity considers how a polynomial behaves as \( x \) becomes extremely large. For \( n > 0 \), \( x^n \) grows indefinitely positive or negative. For \( n < 0 \), \( x^n \) approaches zero, as larger \( x \) values in the negative exponent make the fraction smaller.

Even and Odd Functions

Even and odd functions have distinctive properties based on their symmetry. They play a crucial role in sketching the graph of polynomial functions.

• Even Functions: A function \( f(x) \) is considered even if \( f(-x) = f(x) \) for all \( x \). This means that the graph of the function is symmetric with respect to the y-axis. Think of even powers like \( x^2, x^4 \), where you see a mirrored effect about the y-axis.


• Odd Functions: A function is considered odd if \( f(-x) = -f(x) \) for all \( x \). This denotes symmetry around the origin. For odd powers like \( x^3, x^5 \), the function's graph twists around the origin, appearing like an s-curve.

Recognizing whether a polynomial is even or odd assists in anticipating how its graph will look, simplifying the process of sketching functions on a chart.

Graph Sketching

Graph sketching involves drawing a rough outline of a function based on its equation. It helps us visualize how a function behaves, making it clearer to predict limits and other behaviors.

• When sketching polynomial functions, start by identifying the degree and the leading coefficient. The degree helps determine the overall shape, while the leading coefficient affects the width and direction.


• For functions like \( x^n \), knowing if \( n \) is positive or negative, even or odd, provides insights into the symmetry (discussed earlier) and how the function behaves at infinity.


• A horizontal line indicates a constant function, like \( x^0 \), while v-shaped graphs come from even powers. Odd powers twist around the axes.

Additionally, pay attention to intercepts and any points of inflection where the graph changes its curvature direction. These points are important markers for an accurate sketch.