Question

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 2} \frac{x^{2}+7 x+10}{x^{2}-4 x+4} $$

Short answer

The limit does not exist due to a vertical asymptote at \( x=2 \).

Step-by-step solution

Analyze the Expression

Given the limit \( \lim _{x \rightarrow 2} \frac{x^{2}+7 x+10}{x^{2}-4 x+4} \), we need to determine if the expression is undefined at \( x=2 \). Calculate \( x^2 - 4x + 4 \) when \( x=2 \).\[ (2)^2 - 4 imes 2 + 4 = 4 - 8 + 4 = 0 \] This results in a division by zero, indicating a possible indeterminate form.

Simplify the Expression

Factor the numerator and the denominator to simplify the expression. The numerator \( x^2 + 7x + 10 \) can be factored as \((x+5)(x+2)\). The denominator \( x^2 - 4x + 4 \) can be factored as \((x-2)^2\). So the expression becomes:\[ \frac{(x+5)(x+2)}{(x-2)^2} \].

Determine the Behavior near x=2

Since \( (x-2) \) appears twice in the denominator, and is not a factor of the numerator, the function has a vertical asymptote at \( x = 2 \). This usually indicates that the limit does not exist numerically.

Graphical Analysis

Plot the function \( y = \frac{x^2 + 7x + 10}{x^2 - 4x + 4} \) to observe its behavior around \( x=2 \). The graph will show a vertical asymptote at \( x=2 \), confirming the indication from our simplification that the function diverges.

Conclusion

Since both graphical analysis and algebraic manipulation demonstrate that the function diverges at \( x=2 \) (because of the vertical asymptote), the limit does not exist.

Key concepts

Indeterminate Forms

When dealing with limits, especially as a variable approaches a value that causes a division by zero, we encounter what are known as **indeterminate forms**. In calculus, indeterminate forms often require additional analysis to determine the limit's value, or to conclude if a limit exists at all. In the given exercise, when we substitute \( x = 2 \) in the denominator, the expression results in zero, suggesting an indeterminate form of the type \( \frac{0}{0} \). This indicates that at least one part of the equation needs to be simplified or re-evaluated to determine a limit that can be analyzed meaningfully. When you see such forms, remember they denote that the limit could be anything: finite, infinite, or might not exist. The presence of this form is a signpost that further work is necessary to reveal the nature of the limit.

• Helpful techniques include algebraic simplification, L'Hôpital's rule, or converting the expression into a different form.
• In this problem, simplification helped understand the behavior of the function around the point of interest.

Simplifying Algebraic Expressions

Simplification of algebraic expressions is a crucial step in limit problems, especially when faced with indeterminate forms. Simplifying involves manipulating the expression into a more manageable form, often by factoring or canceling common terms. In this exercise, the numerator \( x^2 + 7x + 10 \) is factored to \((x+5)(x+2)\), and the denominator \( x^2 - 4x + 4 \) is factored to \((x-2)^2\).

By simplifying the expression, we gain insights into how the function behaves near \( x=2 \).

• Factoring helps identify terms that contribute to zero in the numerator and denominator.
• Cancel common factors only if they appear in both the numerator and denominator to simplify the expression.

In this case, the absence of \( (x-2) \) in the numerator despite its presence in the denominator leads to the conclusion that the expression is undefined at that point. This simplification alerts us to the pending vertical asymptote, drawing attention to where and why the function diverges near \( x=2 \). Simplification not only helps solve calculations but also clarifies the function's behavior.

Graphical Analysis of Functions

Graphical analysis provides a visual understanding of how a function behaves, especially useful in cases of indeterminate forms or undefined points. By plotting the graph of the function\( y = \frac{x^2 + 7x + 10}{x^2 - 4x + 4} \), one can visually discern changes at critical points, such as \( x = 2 \). The graph in this exercise shows a distinct vertical line known as a vertical asymptote at \( x=2 \).

This reflects the nature of the limit analyzed algebraically, confirming the analytical deduction that the function diverges as \( x \) approaches 2.

• Asymptotes are lines that the function's graph approaches but never touches.
• Observe how the function trends around points of interest to verify analytical findings.

Graphing allows you to verify algebraic steps and illustrate the function's approach as \( x \) nears the critical point. These visual cues aid in developing intuition about limits and the overall behavior of functions near specific values.