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10. Limits (Computational Techniques)

Dated: 30-10-2024

Theorems

  1. \[\lim[f(x) + g(x)] = \lim f(x) + \lim g(x)\]
  2. \[\lim[f(x) - g(x)] = \lim f(x) - \lim g(x)\]
  3. \[\lim[f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x)\]
  4. \[\lim \left[\frac{f(x)}{g(x)}\right] = \frac{\lim f(x)}{\lim g(x)}\]
  5. \[\lim_{x \rightarrow a}(x^n) = \left(\lim_{x \rightarrow a} (x)\right)^n = a^n\]
  6. \[\lim[k \cdot g(x)] = \lim(k) \cdot \lim g(x) = k \cdot \lim g(x)\]

Limits of Polynomial

Example

\[\lim_{x \rightarrow 5}(x^2 - 4x + 3)\]
\[= \lim_{x \rightarrow 5} x^2 - \lim_{x \rightarrow 5} 4x + \lim_{x \rightarrow 5} 3\]
\[= \lim_{x \rightarrow 5} x^2 - 4 \cdot \lim_{x \rightarrow 5} x + \lim_{x \rightarrow 5} 3\]
\[= 5^2 - 4 \cdot 5 + 3\]
\[= 25 - 20 + 3\]
\[= 8\]

Rules for \(x\) Approaching to \(\infty\)

\[\lim_{x \rightarrow + \infty} \frac{c_0 + c_1 \cdot x^1 + \ldots + c_nx^n}{d_0 + d_1 \cdot x^1 + \ldots + d_nx^n} = \lim_{x \rightarrow + \infty} \frac{c_nx^n}{d_nx^n}\]
\[\lim_{x \rightarrow - \infty} \frac{c_0 + c_1 \cdot x^1 + \ldots + c_nx^n}{d_0 + d_1 \cdot x^1 + \ldots + d_nx^n} = \lim_{x \rightarrow - \infty} \frac{c_nx^n}{d_nx^n}\]

Example

\[\lim_{x \rightarrow - \infty} \frac{4x^2 - x}{2x^3 - 5} = \lim_{x \rightarrow - \infty} \frac{4x^2}{2x^3} = \lim_{x \rightarrow - \infty} \frac{2}{x} = 0\]

References

Read more about notations and symbols.