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Which of the following functions will grow slower? A. lim 2x + 3 x→∞ B. lim 2x² + 2x − 1 x→∞ C. lim 2 + 3 x →∞ Answer: A will grow slower because it is linear

Which of the following functions will grow slower? A. lim 2x + 3 x→∞ B. lim 2x² + 2x − 1 x→∞ C. lim 2 + 3 x →∞ Answer: A will grow slower because it is linear

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Problem 2 - Identify the Error(s) and Solve Correctly #### Question: Consider the following functions and determine which one will grow slower as \(x\) approaches infinity: A. \(\lim_{{x \to \infty}} (2x + 3)\) B. \(\lim_{{x \to \infty}} (2x^2 + 2x - 1)\) C. \(\lim_{{x \to \infty}} (2^x + 3)\) #### Answer: Option **A** will grow slower because it represents a linear function. ### Explanation: 1. **Option A:** \(\lim_{{x \to \infty}} (2x + 3)\) - This is a linear function. As \(x\) approaches infinity, the term \(2x\) dominates over the constant term 3. Linear functions grow at a steady rate. 2. **Option B:** \(\lim_{{x \to \infty}} (2x^2 + 2x - 1)\) - This is a quadratic function. As \(x\) approaches infinity, the \(x^2\) term dominates, causing the function to grow much faster than a linear function. 3. **Option C:** \(\lim_{{x \to \infty}} (2^x + 3)\) - This is an exponential function. As \(x\) approaches infinity, the term \(2^x\) grows exponentially, much faster than both linear and quadratic functions. Linear functions indeed grow slower compared to quadratic or exponential functions as the variable \(x\) approaches infinity. Therefore, the correct choice is Option **A**.
Transcribed Image Text:### Problem 2 - Identify the Error(s) and Solve Correctly #### Question: Consider the following functions and determine which one will grow slower as \(x\) approaches infinity: A. \(\lim_{{x \to \infty}} (2x + 3)\) B. \(\lim_{{x \to \infty}} (2x^2 + 2x - 1)\) C. \(\lim_{{x \to \infty}} (2^x + 3)\) #### Answer: Option **A** will grow slower because it represents a linear function. ### Explanation: 1. **Option A:** \(\lim_{{x \to \infty}} (2x + 3)\) - This is a linear function. As \(x\) approaches infinity, the term \(2x\) dominates over the constant term 3. Linear functions grow at a steady rate. 2. **Option B:** \(\lim_{{x \to \infty}} (2x^2 + 2x - 1)\) - This is a quadratic function. As \(x\) approaches infinity, the \(x^2\) term dominates, causing the function to grow much faster than a linear function. 3. **Option C:** \(\lim_{{x \to \infty}} (2^x + 3)\) - This is an exponential function. As \(x\) approaches infinity, the term \(2^x\) grows exponentially, much faster than both linear and quadratic functions. Linear functions indeed grow slower compared to quadratic or exponential functions as the variable \(x\) approaches infinity. Therefore, the correct choice is Option **A**.
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