Problem 53 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ g(x)=x^{2}-5 ; \quad \text { find } \lim _{x \rightarrow 0} g(x) \text { and } \lim _{x \rightarrow-1} g(x). $$

Step-by-Step Solution

Answer

\( \lim_{x \to 0} g(x) = -5 \) and \( \lim_{x \to -1} g(x) = -4 \).

Step 1: Understand the Function

The given function is a quadratic function, expressed as \( g(x) = x^2 - 5 \). Recognize that this function describes a parabola opening upwards, with its vertex at (0, -5).

Step 2: Analyze the Limit as \( x \to 0 \)

To find \( \lim_{x \to 0} g(x) \), substitute \( x = 0 \) into \( g(x) \):\[g(0) = 0^2 - 5 = -5.\]Since \( g(x) \) is continuous at \( x = 0 \), the limit exists, and \( \lim_{x \to 0} g(x) = -5 \).

Step 3: Analyze the Limit as \( x \to -1 \)

To find \( \lim_{x \to -1} g(x) \), substitute \( x = -1 \) into \( g(x) \):\[g(-1) = (-1)^2 - 5 = 1 - 5 = -4.\]Since \( g(x) \) is continuous at \( x = -1 \), the limit exists, and \( \lim_{x \to -1} g(x) = -4 \).

Step 4: Graph the Function

Graph the function \( g(x) = x^2 - 5 \). The function is a parabola that opens upwards with a vertex at the point (0, -5). This supports the calculations as \( g(x) \) approaches -5 as \( x \to 0 \) and -4 as \( x \to -1 \).

Key Concepts

Quadratic Functions

A quadratic function is a type of polynomial function characterized by its highest power of the variable being 2. These functions can be represented by the standard form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the equation \( g(x) = x^2 - 5 \), we specifically observe a simplified version where \( a=1 \), \( b=0 \), and \( c=-5 \).

• The term \( x^2 \) indicates that the function is indeed quadratic.
• The constant \(-5\) shifts the entire graph downward by 5 units compared to the basic parabola \( y = x^2 \).
• This function, therefore, describes a U-shaped curve that opens upward since \( a \) is positive.

Understanding quadratics is essential because they model various real-world phenomena, from projectile motion to optimizing areas or costs in business. The vertex form of a quadratic, \( y = a(x-h)^2+k \), reveals the vertex \((h, k)\) directly. Here, the vertex is at \((0, -5)\), meaning the minimum point of this function is located there.

Continuous Functions

Continuous functions are an essential concept in calculus. A function is continuous at a point if there is no interruption in its graph at that point. For a function to be continuous at \( x = a \), three conditions must be satisfied:

• The function \( f(a) \) is defined.
• The limit \( \lim_{x \to a} f(x) \) exists.
• \( \lim_{x \to a} f(x) = f(a) \).

With the function \( g(x) = x^2 - 5 \), continuity can be verified at any value of \( x \), as this polynomial function does not have any points of discontinuity. When evaluating the limit as \( x \to 0 \), we find that it equals the value of the function at that point, which is \(-5\). Similarly, as \( x \to -1 \), the limit results match the function value of \(-4\). This property of continuity is crucial as it assures us that the function behavior is predictable and smooth around these points.

Graphing Parabolas

Graphing parabolas involves drawing the curve of a quadratic function on a coordinate plane. The graph of a quadratic function, such as \( g(x) = x^2 - 5 \), is known as a parabola. Here are key steps to consider when graphing:

• Identify the Vertex: For a function in the form \( ax^2 + bx + c \), the vertex can be determined using \( h = -\frac{b}{2a} \) and \( k = g(h) \). For our function, the vertex is \((0, -5)\).
• Determine the Axis of Symmetry: This is a vertical line passing through the vertex, so for \( g(x) = x^2 - 5 \), it is \( x = 0 \).
• Identify the Direction: Since \( a > 0 \) in the expression \( ax^2 \), the parabola opens upward.
• Plot Additional Points: Choosing other values for \( x \) helps in sketching a more accurate curve. For example, using points \((-1, -4)\) and \((1, -4)\), we can see the parabola's symmetric nature.

This structured approach aids in graphing the parabola accurately, ensuring a visual understanding of how the quadratic function behaves across all values of \( x \). Knowing how to graph these functions can be incredibly beneficial for analyzing their properties and determining key factors like intercepts and vertex positions.