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Chapter 1: Problem 1

In Problems \(1-6\), find the indicated limit. $$ \lim _{x \rightarrow 3}(x-5) $$

Short Answer

Expert verified
The limit is -2.

Step by step solution

01

Substitute the Limit Value

To find the limit of \( f(x) = x - 5 \) as \( x \) approaches 3, we start by directly substituting \( x = 3 \) into the function: \( f(3) = 3 - 5. \)
02

Simplify the Expression

Simplify the expression obtained after substitution: \( f(3) = 3 - 5 = -2. \) This is the value that \( f(x) \) approaches as \( x \) approaches 3.

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Key Concepts

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

Substitution Method
The substitution method is a straightforward technique used to find the limit of a function. When using this method, we simply replace the variable in the function with the value it is approaching. For example, to find \( \lim_{x \rightarrow 3}(x-5) \), we substitute \( x = 3 \) directly into the function \( f(x) = x - 5 \).
This step involves evaluating \( f(3) \), which means calculating \( 3 - 5 \).
There are a few key things to remember when using the substitution method:
  • Ensure the function is defined at the point of substitution.
  • If the function is not defined, or results in an indeterminate form (like \(\frac{0}{0}\)), consider other methods such as factoring or L'H么pital's rule.
  • Substitution works best for simple polynomial functions or functions where plugging in the value doesn鈥檛 create any discontinuity.
By substituting the approach value into the function, we can often obtain the limit directly in one step.
Evaluating Limits
Evaluating limits is a fundamental aspect of calculus where we determine what value a function approaches as the variable gets closer to a certain point. In our example, we evaluated the limit of \( x - 5 \) as \( x \) approaches 3.
When evaluating limits, here are a few steps to guide you:
  • First, try direct substitution into the function.
  • If the direct substitution yields an undefined value, seek alternative paths such as factoring, rationalizing, or applying advanced techniques like L'H么pital's rule.
  • Reassess if the function is continuous at the point of approach; if it is continuous, the function's value at that point will equal the limit.
In simple linear functions like the one presented, direct substitution works perfectly, and direct evaluation gives the correct result. This makes direct substitution a powerful first step when evaluating limits.
Simplifying Expressions
Simplifying expressions is the process of reducing them to their simplest form. Simplification makes evaluation more straightforward and helps in understanding the behavior of a function. After substitution, our goal is to simplify the resulting expression to find the limit.
Consider our example where we needed to simplify \( 3 - 5 \) after substituting in the value:
  • Always perform basic arithmetic operations like addition, subtraction, multiplication, and division carefully.
  • After substituting, simplify the expression step by step.
  • Check if any further simplification is possible, like factoring or canceling terms, especially for complex expressions.
In our example, simplifying meant performing the straightforward calculation to yield \( -2 \), which was the limit. Simplification is crucial as it leads to uncovering the precise behavior of the function at a particular point.

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Most popular questions from this chapter

In Problems \(41-48\), determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ g(x)=\left\\{\begin{array}{cl} \frac{\sin x}{x}, & x \neq 0 \\ 0, & x=0 \end{array}\right. $$

In Problems \(1-22,\) find the indicated limit or state that it does not exist. $$ \lim _{t \rightarrow 2^{-}}([t]-t) $$

In Problems \(31-36,\) find the equations of all vertical and horizontal asymptotes for the given function. $$ F(x)=\frac{x^{2}}{x^{2}-1} $$

$$ \text { Prove using an } \varepsilon-\delta \text { argument that } \lim _{x \rightarrow 3}(2 x+1)=7 \text { . } $$

In Problems \(1-15,\) state whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(t)=\left\\{\begin{array}{cl} t^{2}-9 & \text { if } t \leq 3 \\\ (3-t)^{2} & \text { if } t>3 \end{array}\right. $$
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