Solution: To find vertical asymptotes and holes, we examine the denominator of $ D(t) = rac{t^2 + 1}{t - 2} $.
["# Understanding Vertical Asymptotes and Holes: Analyzing the Denominator in Rational Functions", "When studying rational functions like\n$$\nD(t) = \frac{t^2 + 1}{t - 2},\n$$\na key step in understanding the function’s behavior is identifying vertical asymptotes and holes. These features are primarily determined by examining the denominator, making it essential to analyze where the function is undefined.", "## Why the Denominator Matters", "In rational expressions, vertical asymptotes occur at values of $ t $ that make the denominator equal to zero, provided these values do not cancel out with the numerator. If the denominator becomes zero without being canceled, the function “blows up” — resulting in a vertical asymptote.", "Conversely, holes appear when both the numerator and denominator share a common factor — meaning the function is undefined at that point, but the discontinuity is removable. However, in the given function $ D(t) = \dfrac{t^2 + 1}{t - 2} $, the numerator $ t^2 + 1 $ has no real roots (since $ t^2 + 1 = 0 $ implies $ t = \pm i $, which are imaginary), so there are no holes.", "## Step-by-Step: Finding Vertical Asymptotes", "To locate vertical asymptotes:", "1. Set the denominator equal to zero:\n $$\n t - 2 = 0\n $$\n Solving gives $ t = 2 $.", "2. Check if the numerator is non-zero at this value:\n $$\n t^2 + 1 \quad \ ext{at} \quad t = 2 \Rightarrow 2^2 + 1 = 5 <br/>\neq 0\n $$\n Since the numerator is not zero when $ t = 2 $, the function is undefined there, and because no factor cancels out, there is a vertical asymptote at $ t = 2 $.", "## Summary", "- Denominator Analysis: $ t - 2 = 0 $ ⇒ $ t = 2 $ — a candidate for a vertical asymptote.\n- Numerator Check: $ t^2 + 1 $ is non-zero at $ t = 2 $ — no cancellation, so function remains undefined.\n- Conclusion: There is a vertical asymptote at $ t = 2 $. There are no holes because the numerator does not share any factors with the denominator.", "## Why This Matters", "Understanding vertical asymptotes helps in graphing rational functions accurately and assessing behavior near critical points. Though $ t^2 + 1 $ never equals zero for real $ t $, and doesn’t cancel with $ t - 2 $, always examine the denominator first — it reveals the fundamental undefined points that define asymptotes and discontinuities.", "Debug tip: If you see no denominator zero or common factor with numerator, skip asymptotic analysis — your function is smooth (except possibly at domain boundaries).", "---", "Keywords: vertical asymptote, holes in rational functions, denominator analysis, $ D(t) = \frac{t^2 + 1}{t - 2} $, rational function asymptotes, where is $ D(t) $ undefined, simplifying rational functions, $ t^2 + 1 $, function discontinuities.", "Optimize your study of rational functions by always inspecting the denominator first — it holds the key to unlocking asymptotes and discontinuities!"]
