Solve a problem

LimitsCalculusCollegeabout 2:14

Find lim x->3 (x^2 + x - 12)/(x^2 - 9): animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 7/6 ≈ 1.166666667.

SpeedLanguage

Chapters

Transcript

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 (0.8 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\lim_{x \to 3} \frac{{x}^{2}+ x-12}{{x}^{2}-9}$
2. Step by step
1. $\lim_{x \to 3} \frac{{x}^{2}+ x-12}{{x}^{2}-9} = \lim_{x \to 3} \frac{2x+1}{2x}$
  Indeterminate form 0/0 — apply L'Hôpital's rule
2. $\lim_{x \to 3} \frac{2x+1}{2x} = \frac{2\cdot3+1}{2\cdot3} = \frac{7}{6} \approx 1.166666667$
  Direct substitution
3. The answer
$\lim_{x \to 3} \frac{{x}^{2}+ x-12}{{x}^{2}-9} = \frac{7}{6} \approx 1.166666667$
lim x→3 (x² + x - 12) / (x² - 9) = 7/6 ≈ 1.166666667
4. Check
- Numerical values from the right agree: f(3.01) = 1.1663894, f(3.001) = 1.1666389, f(3.0001) = 1.1666639, f(3.00001) = 1.1666664 ✓
- Numerical values from the left agree: f(2.99) = 1.1669449, f(2.999) = 1.1666944, f(2.9999) = 1.1666694, f(2.99999) = 1.1666669 ✓

Transcript

The problem. Here is the problem: (x squared plus x minus 12)/(x squared minus 9). Find the limit of (x squared plus x minus 12) / (x squared minus 9) as x → 3.

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). Direct substitution. The expression is continuous at x equals 3, so the limit equals its value there.

The answer. So the answer is Limit equals 7 over 6 approximately 1.166666667. 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(3.01) equals 1.1663894, f(3.001) equals 1.1666389, f(3.0001) equals 1.1666639, f(3.00001) equals 1.1666664, which checks out. Numerical values from the left agree: f(2.99) equals 1.1669449, f(2.999) equals 1.1666944, f(2.9999) equals 1.1666694, f(2.99999) equals 1.1666669, which checks out.

About limits

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

Find lim x->3 (x^2 + x - 12)/(x^2 - 9): Limits Lesson