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Chapter 1: Problem 16

Find the limit. $$ \lim _{x \rightarrow 3} \frac{2 x-3}{x+5} $$

Short Answer

Expert verified
The limit of \(\frac{2x-3}{x+5}\) as \(x\) approaches 3 is \(\frac{3}{8}\).

Step by step solution

01

Substitute the Value

Start by substituting the given value that \(x\) is approaching (which is 3), into the equation \(\frac{2x-3}{x+5}\).
02

Perform the Calculation

Calculate the expression obtained after substitution: \(\frac{2*3-3}{3+5} = \frac{3}{8}\).
03

State the Result

State that the limit of \(\frac{2x-3}{x+5}\) as \(x\) approaches 3 is \(\frac{3}{8}\).

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Key Concepts

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

Evaluating Limits
Evaluating limits is an essential concept in calculus. When we talk about limits, we are trying to understand what value a function approaches as the input approaches a certain number. It's like predicting where a car is headed when you know its path.

To evaluate limits, we often follow some standardized steps to ensure accuracy. These include:
  • Identify the point the variable is approaching. In our example, this would be where \( x \rightarrow 3 \).
  • Substitute the value, if feasible, to see the result of the function.
  • Analyze any form of indeterminacy, like \( \frac{0}{0} \), which might require more advanced techniques like factoring or rationalizing.
Through this process, the goal is to find the exact value that a function trends towards without necessarily reaching it. In simple terms, it's the behavior of the function as the input gets close to a specific value.
Substitution Method
The substitution method for evaluating limits is one of the most straightforward techniques when the function allows for it. This involves directly plugging in the value of \( x \) that it's approaching into the function.

This method is used when the function is well-behaved at the point you're examining. Here's how it works:
  • Take the limit expression and replace the variable \( x \) with the number it approaches.
  • Calculate the result. For example, in evaluating \( \lim_{x \rightarrow 3} \frac{2x-3}{x+5} \), substituting directly gives \( \frac{2\cdot3 - 3}{3 + 5} \).
  • If a simple evaluation like this doesn't cause undefined expressions (like division by zero), the substitution is valid.
Using substitution helps make quick work of finding limits and can confirm whether more complex methods are necessary.
Continuous Functions
Continuous functions play a crucial role when evaluating limits. A continuous function, in basic terms, is one where small changes in input lead to small changes in output, with no breaks or jumps.

Understanding the nature of continuous functions can greatly simplify limit problems since for any continuous function at a point, the limit as \( x \) approaches \( c \) is simply the value of the function at \( c \). This makes finding the limit straightforward.
  • Continuity implies that you can apply the substitution method without worry of undefined operations.
  • Most polynomial, exponential, sinusoidal, and other basic functions are continuous over their domains.
  • In calculus problems, if you know a function is continuous at a point, you can directly substitute to find the limit.
Working with continuous functions ensures the process of evaluating limits is smooth, as their nature avoids discontinuities that can lead to more complex calculations.

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Most popular questions from this chapter

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 1} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} x^{3}+1, & x<1 \\ x+1, & x \geq 1 \end{array}\right. $$

Describe the interval(s) on which the function is continuous. $$ f(x)=\sec \frac{\pi x}{4} $$

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow \pi / 2} \sec x $$

Describe how the functions \(f(x)=3+\llbracket x \rrbracket\) and \(g(x)=3-\llbracket-x \rrbracket\) differ.

Use a graphing utility to graph the function. From the graph, estimate \(\lim _{x \rightarrow 0^{+}} f(x) \quad\) and \(\lim _{x \rightarrow 0^{-}} f(x)\) Is the function continuous on the entire real line? Explain. $$ f(x)=\frac{\left|x^{2}-4\right| x}{x+2} $$
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