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Find lim x->9 (sqrt(x) - 3)/(x - 9): animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 1/6 ≈ 0.1666666667.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.9 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\lim_{x \to 9} \frac{\sqrt{x}-3}{x-9}$
2. Step by step
1. $\lim_{x \to 9} \frac{\sqrt{x}-3}{x-9} = \lim_{x \to 9} \frac{\frac{\frac{1}{2}}{\sqrt{x}}}{1}$
  Indeterminate form 0/0 — apply L'Hôpital's rule
2. $\lim_{x \to 9} \frac{\frac{\frac{1}{2}}{\sqrt{x}}}{1} = \lim_{x \to 9} \frac{\frac{1}{2}}{\sqrt{x}}$
  Simplify
3. $\lim_{x \to 9} \frac{\frac{1}{2}}{\sqrt{x}} = \frac{\frac{1}{2}}{\sqrt{9}} = \frac{1}{6} \approx 0.1666666667$
  Direct substitution
3. The answer
$\lim_{x \to 9} \frac{\sqrt{x}-3}{x-9} = \frac{1}{6} \approx 0.1666666667$
lim x→9 (sqrt(x) - 3) / (x - 9) = 1/6 ≈ 0.1666666667
4. Check
- Numerical values from the right agree: f(9.01) = 0.1666204, f(9.001) = 0.16666204, f(9.0001) = 0.1666662, f(9.00001) = 0.16666662 ✓
- Numerical values from the left agree: f(8.99) = 0.16671299, f(8.999) = 0.1666713, f(8.9999) = 0.16666713, f(8.99999) = 0.16666671 ✓

Transcript

The problem. Here is the problem: (sqrt(x) minus 3)/(x minus 9). Find the limit of (sqrt(x) minus 3) / (x minus 9) as x → 9.

Step by step. Indeterminate form 0 over 0 — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule). Simplify. Direct substitution. The expression is continuous at x equals 9, so the limit equals its value there.

The answer. So the answer is Limit equals 1 over 6 approximately 0.1666666667. A limit describes the value a function approaches, not necessarily the value it takes. When substitution gives an indeterminate form, the expression is rewritten (L prime Hôpital prime s rule, a conjugate, or logarithms) until substitution works.

Check. Numerical values from the right agree: f(9.01) equals 0.1666204, f(9.001) equals 0.16666204, f(9.0001) equals 0.1666662, f(9.00001) equals 0.16666662, which checks out. Numerical values from the left agree: f(8.99) equals 0.16671299, f(8.999) equals 0.1666713, f(8.9999) equals 0.16666713, f(8.99999) equals 0.16666671, which checks out.

About limits

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

Find lim x->9 (sqrt(x) - 3)/(x - 9): Limits Lesson