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LimitsCalculusCollegeabout 1:50

Find lim x->0 1/x^2: animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = ∞.

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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 (0.7 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\lim_{x \to 0} \frac{1}{{x}^{2}}$
2. Step by step
1. Nonzero numerator over a denominator approaching 0
2. Conclusion
3. The answer
$\lim_{x \to 0} \frac{1}{{x}^{2}} = \infty$
lim x→0 1 / x² = ∞
4. Check
- Numerical values from the right agree: f(0.01) = 10000, f(0.001) = 1000000, f(0.0001) = 100000000, f(0.00001) = 10000000000 ✓
- Numerical values from the left agree: f(-0.01) = 10000, f(-0.001) = 1000000, f(-0.0001) = 100000000, f(-0.00001) = 10000000000 ✓

Transcript

The problem. Here is the problem: 1/x squared. Find the limit of 1 / x squared as x → 0.

Step by step. Nonzero numerator over a denominator approaching 0. The numerator approaches 1 while the denominator approaches 0, so the quotient grows without bound. Its sign on each side decides +∞ or −∞: from the right the quotient → ∞; from the left the quotient → ∞. Conclusion. The function is unbounded, so the limit is +∞ (strictly, the limit does not exist as a real number; it diverges to +∞).

The answer. So the answer is Limit equals ∞. 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(0.01) equals 10000, f(0.001) equals 1000000, f(0.0001) equals 100000000, f(0.00001) equals 10000000000, which checks out. Numerical values from the left agree: f(negative 0.01) equals 10000, f(negative 0.001) equals 1000000, f(negative 0.0001) equals 100000000, f(negative 0.00001) equals 10000000000, which checks out.

About limits

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

Find lim x->0 1/x^2: Animated Limits Lesson, Step by Step