As \( x \) approaches 1 from the right, \( x - 1 \) approaches 0 from the positive side, making \( f(x) = rac{3x + 2}{x - 1} \) approach \( +\infty \).

["Understanding the Behavior of ( f(x) = \dfrac{3x + 2}{x - 1} ) as ( x \ o 1^+ )", "When analyzing functions near key points like ( x = 1 ), understanding limits helps reveal important behavior such as asymptotes, continuity, and undefined regions. In this article, we focus on the one-sided limit of the function ( f(x) = \dfrac{3x + 2}{x - 1} ) as ( x ) approaches 1 from the right (( x \ o 1^+ )).", "---", "### What Does It Mean for ( x - 1 ) to Approach 0 from the Positive Side?", "As ( x ) approaches 1 from the right (( x \ o 1^+ )), the value of ( x - 1 ) becomes a small positive number—greater than 0 but very close to 0. For example:\n- If ( x = 1.001 ), then ( x - 1 = 0.001 > 0 )\n- If ( x = 1.000001 ), then ( x - 1 = 0.000001 > 0 )", "This means ( x - 1 ) approaches 0 from the positive side, also written as:\n[\n\lim_{x \ o 1^+} (x - 1) = 0^+\n]", "---", "### How Does This Affect the Value of ( f(x) = \dfrac{3x + 2}{x - 1} )?", "The numerator of ( f(x) ), ( 3x + 2 ), is continuous and plays a stable role:\nAt ( x = 1 ), it evaluates to ( 3(1) + 2 = 5 ).\nSo the expression becomes:\n[\nf(x) = \dfrac{5}{\ ext{a very small positive number}}\n]", "As the denominator approaches 0 from the positive side, the entire fraction grows without bound in the positive direction.", "Mathematically, we say:\n[\n\lim_{x \ o 1^+} f(x) = +\infty\n]", "---", "### Graphical and Practical Implications", "When graphing ( f(x) = \dfrac{3x + 2}{x - 1} ), this limit reveals a vertical asymptote at ( x = 1 ) from the right. On the graph, the function shoots upward toward positive infinity as ( x ) gets closer to 1, never actually reaching ( +\infty ), but becoming larger and larger.", "This behavior is essential for:\n- Identifying asymptotes\n- Predicting function behavior near critical points\n- Understanding domain restrictions", "---", "### Final Summary", "- As ( x \ o 1^+ ), ( x - 1 \ o 0^+ ), a small positive number.\n- The numerator ( 3x + 2 ) approaches 5.\n- Thus, ( f(x) = \dfrac{3x + 2}{x - 1} \ o +\infty ).\n- This defines a vertical asymptote at ( x = 1 ) from the right.", "---", "Key Takeaway: Studying one-sided limits near undefined points helps clarify function behavior and long-term trends—crucial knowledge in calculus and real-world modeling.", "---", "Keywords: limit as ( x \ o 1^+ ), ( f(x) = \frac{3x + 2}{x - 1} ), ( x - 1 \ o 0^+ ), ( f(x) \ o +\infty ), vertical asymptote, calculus, function behavior.", "---", "Read More:\n- How to Compute One-Sided Limits\n- Vertical Asymptotes and Rational Functions\n- Understanding Infinite Limits in Calculus", "---", "Understanding ( f(x) ) as ( x \ o 1^+ ) proves that subtle changes in input near critical points can lead to dramatic outcomes, a fundamental concept in mathematical analysis."]