Question

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 1} x^{2}+3 x-5 $$

Short answer

The limit is \(-1\).

Step-by-step solution

Analyze the Function

The given function is \( f(x) = x^2 + 3x - 5 \). We need to find the limit of this function as \( x \) approaches 1. The limit expression is \( \lim _{x \rightarrow 1} (x^2 + 3x - 5) \).

Substitute x = 1 into the Function

Since \( f(x) = x^2 + 3x - 5 \) is continuous at \( x = 1 \), we can directly substitute \( x = 1 \) into the function. Calculate: \( 1^2 + 3 \cdot1 - 5 = 1 + 3 - 5 \).

Simplify the Expression

Simplifying the expression from Step 2, we get: \( 1 + 3 - 5 = -1 \).

Graphically Analyze the Limit

Create a graph of the function \( f(x) = x^2 + 3x - 5 \). Observe the behavior of the graph near \( x = 1 \). The y-value of the graph at \( x = 1 \) should match the limit calculated.

Numerically Approximate the Limit

Choose values of \( x \) close to 1, such as 0.9 and 1.1, and calculate \( f(x) \). For example, \( f(0.9) = (0.9)^2 + 3 \times 0.9 - 5 = -1.01 \) and \( f(1.1) = (1.1)^2 + 3 \times 1.1 - 5 = -0.99 \). These values are close to -1, confirming our limit value.

Key concepts

Continuity

Continuity is a fundamental concept in calculus that describes a situation where a function behaves predictably, without any sudden jumps, breaks, or holes at a given point. When we say a function is continuous at a specific point, such as \( x = 1 \), it means the limit of the function as \( x \) approaches that point is exactly equal to the function's value at that point.
For our function, \( f(x) = x^2 + 3x - 5 \), we found it to be continuous at \( x = 1 \). This implies that not only does the limit \( \lim_{x \to 1} (x^2 + 3x - 5) \) exist, but it also equals \( f(1) \). When a function is continuous, evaluating the limit becomes straightforward: substitute the point directly into the function.
Continuity simplifies many calculus problems because it enables direct substitution. Without continuity, we may need alternative methods to find the limit, such as limit laws or L'Hôpital's rule.

Numerical Approximation

Numerical approximation is a practical technique used when an exact limit is difficult or impossible to calculate directly. This approach helps us verify calculated limits or explore the behavior of a function as the variable approaches a specific value.
For our example, we chose values near \( x = 1 \), specifically \( x = 0.9 \) and \( x = 1.1 \), to approximate the limit of \( f(x) = x^2 + 3x - 5 \).

• Calculating \( f(0.9) \) yields approximately \(-1.01\).
• Calculating \( f(1.1) \) yields approximately \(-0.99\).

Both results suggest that the limit is close to \(-1\), matching the value obtained by direct substitution. Using numerical approximation enables verification of our answer and provides confidence about the continuity and behavior of the function around \( x = 1 \).
Employing numerical approximation is handy for complex functions where traditional algebraic solutions are cumbersome.

Graphical Analysis

Graphical analysis of functions involves sketching or using graphing tools to visualize the behavior of a function near a particular point. It's an excellent way of understanding limits and verifying numerical or analytical results.
To analyze \( f(x) = x^2 + 3x - 5 \) graphically, we plot the function and observe its behavior as \( x \) approaches 1. When you see the graph, focus on the direction of the curve as \( x \) is close to 1:

• The graph smoothly transitions through \( x = 1 \) without any abrupt changes, which confirms continuity.
• The value of \( f(x) \) at \( x = 1 \), i.e., the y-coordinate, aligns with our calculated limit, \(-1\).

Graphical representations reinforce understanding by providing a visual check for algebraic calculations. They also help illustrate how smoothly or abruptly function values change, further confirming mathematical concepts such as continuity.