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Chapter 1: Problem 40

Find the limit. \(\lim _{x \rightarrow 3} \frac{x-2}{x^{2}}\)

Short Answer

Expert verified
\(\frac{1}{9}\)

Step by step solution

01

Compute the function value at the limit

When \(x=3\), the function value is \(\frac{3-2}{3^{2}}=\frac{1}{9}\)
02

Apply limit rule

The limit of a function as x approaches a is the function value at a if the function is defined at a. In this case, the function is defined at 3, and the function value at 3 is \(\frac{1}{9}\). Thus, the limit of the function as x approaches 3 is \(\frac{1}{9}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a vast area of mathematics that deals with change and motion. At its core, it consists of two branches: differential calculus, which concerns the rate of change of quantities (known as derivatives), and integral calculus, which focuses on the accumulation of quantities (known as integrals). One of the foundational concepts in differential calculus is the limit. A limit captures the behavior of a function as we approach a certain input value and is crucial in defining derivatives and continuous functions. Understanding limits is also essential in evaluating the behavior of functions at points where they may not be explicitly defined or may exhibit indeterminate forms.
Limit Evaluation
Evaluating a limit means finding the value a function approaches as the input (or the variable) gets closer to a specific point. One basic approach to finding limits is by directly substituting the point into the function, provided the function is defined at that point and is continuous. If the direct substitution leads to an indeterminate form, other techniques such as factoring, rationalizing, or applying L鈥橦么pital's rule may be used. Additionally, understanding the behavior of a function near the point of interest is useful when dealing with limits at infinity or limits that involve undefined expressions such as division by zero.
Rational Functions
Rational functions are ratios of two polynomials. They are expressed in the form \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not zero. The points where the denominator \( Q(x) \) is zero are known as discontinuities or poles of the function. The behavior of rational functions near these points or as \( x \) approaches infinity can be particularly interesting and complex. To find a limit involving a rational function, it's important to consider factors such as the degree of the polynomials in the numerator and denominator, which can give insight into the end behavior of the function.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=g(x)\) for all real numbers other than \(x=0\), and \(\lim _{x \rightarrow 0} f(x)=L, \quad\) then \(\quad \lim _{x \rightarrow 0} g(x)=L\)

Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\frac{|x+2|}{x+2} $$

Consider \(f(x)=\frac{\sec x-1}{x^{2}}\). (a) Find the domain of \(f\). (b) Use a graphing utility to graph \(f .\) Is the domain of \(f\) obvious from the graph? If not, explain. (c) Use the graph of \(f\) to approximate \(\lim _{x \rightarrow 0} f(x)\). (d) Confirm the answer in part (c) analytically.

Find the constant \(a\), or the constants \(a\) and \(b\), such that the function is continuous on the entire real line. $$ f(x)=\left\\{\begin{array}{ll} 2, & x \leq-1 \\ a x+b, & -1

Discuss the continuity of each function. $$ f(x)=\frac{1}{x^{2}-4} $$
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