Estimating Limits from Tables

Step 1: Examine values as x approaches 2 from the left: x: 1.9 → 1.99 → 1.999 → 2 f(x): 3.8 → 3.98 → 3.998 → ? Values approach 4

Step 2: Examine values as x approaches 2 from the right: x: 2 ← 2.001 ← 2.01 ← 2.1 f(x): ? ← 4.002 ← 4.02 ← 4.2 Values approach 4

Step 3: Check agreement: Left-hand limit ≈ 4 Right-hand limit ≈ 4 Both agree!

Step 4: Conclusion: lim(x→2) f(x) = 4

Answer: 4

Step 1: Examine from the right (positive x): x: 0.1 → 0.01 → 0.001 → 0 f(x): 0.9983 → 0.99998 → 0.999999 → ? Approaching 1

Step 2: Examine from the left (negative x): x: -0.1 → -0.01 → -0.001 → 0 f(x): 0.9983 → 0.99998 → 0.999999 → ? Also approaching 1

Step 3: Verify symmetry: Function values are the same from both sides Both approach 1

Step 4: Estimate the limit: lim(x→0) (sin(x)/x) ≈ 1

Step 5: Note: This is a famous limit in calculus! Exact value is 1

Answer: 1

Step 1: Create a table approaching from the left: x: 0.9 0.99 0.999 x³-1: -0.271 -0.0299 -0.003 x-1: -0.1 -0.01 -0.001 f(x): 2.71 2.9701 2.997

Step 2: Create a table approaching from the right: x: 1.1 1.01 1.001 x³-1: 0.331 0.0303 0.003 x-1: 0.1 0.01 0.001 f(x): 3.31 3.0301 3.003

Step 3: Analyze the pattern: From left: 2.71 → 2.9701 → 2.997 → ? From right: 3.31 → 3.0301 → 3.003 → ? Both approaching 3

Step 4: Estimate: lim(x→1) (x³ - 1)/(x - 1) ≈ 3

Step 5: Verification (algebraic): x³ - 1 = (x - 1)(x² + x + 1) (x³ - 1)/(x - 1) = x² + x + 1 At x = 1: 1 + 1 + 1 = 3 ✓

Answer: 3

Step 1: Examine the pattern: As x gets smaller (closer to 0): x: 0.1 → 0.01 → 0.001 → 0.0001 → 0⁺ f(x): 10 → 100 → 1000 → 10000 → ?

Step 2: Observe the behavior: The function values are growing without bound They're increasing toward infinity

Step 3: Conclusion: The limit does not exist as a finite number We say: lim(x→0⁺) (1/x) = ∞

Step 4: Important note: This is an "infinite limit" ∞ is not a number, it describes unbounded growth

Step 5: Graphical interpretation: The graph has a vertical asymptote at x = 0

Answer: The limit is ∞ (infinite limit)