Find lim x->infinity (5x^3 - x)/(2x^3 + 7): animated lesson

1. 1. The problem
   Worked example

   $\lim_{x \to \infty} \frac{5{x}^{3}- x}{2{x}^{3}+7}$
2. 2. Step by step
   1. $$\lim_{x \to \infty} \frac{5{x}^{3}- x}{2{x}^{3}+7} = \lim_{x \to \infty} \frac{15{x}^{2}-1}{6{x}^{2}}$$
      Indeterminate form ∞/∞ — apply L'Hôpital's rule
   2. $$\lim_{x \to \infty} \frac{15{x}^{2}-1}{6{x}^{2}} = \lim_{x \to \infty} \frac{30x}{12x}$$
      Indeterminate form ∞/∞ — apply L'Hôpital's rule
   3. $$\lim_{x \to \infty} \frac{30x}{12x} = \frac{5}{2}$$
      Simplify
   4. $$\lim_{x \to \infty} \frac{5}{2} = \frac{5}{2}$$
      The expression is constant
3. 3. The answer
   $$\lim_{x \to \infty} \frac{5{x}^{3}- x}{2{x}^{3}+7} = \frac{5}{2} \approx 2.5$$

   lim x→∞ (5 x³ - x) / (2 x³ + 7) = 5/2 ≈ 2.5

4. 4. Check
   • ✓Numerical values as x grows agree: f(127.18281) = 2.4999648, f(1271.8281) = 2.4999997, f(12718.281) = 2.5, f(127182.81) = 2.5 ✓

Evaluate limits by substitution, factoring, standard limits and L'Hôpital's rule.

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