5. Indeterminate Form Resolver — Theory
An indeterminate form occurs when direct substitution into a limit produces an expression like
\(\frac{0}{0}\) or \(\infty-\infty\). These do not determine the limit by themselves — you must
rewrite the expression into a form whose limit can be evaluated.
\[
\text{Common indeterminate forms: }
\frac{0}{0},\;\frac{\infty}{\infty},\;0\cdot\infty,\;\infty-\infty,\;1^{\infty},\;0^{0},\;\infty^{0}.
\]
1) The core idea: rewrite first
The goal is to transform the expression into something standard, such as a removable factor cancellation,
a known trig limit, or a product that becomes a clean fraction.
2) \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\)
- Factor and cancel common factors (typical in rational functions).
- Rationalize using the conjugate when square roots appear.
- Use known limits and equivalents near 0 (e.g. \(\sin x\sim x\)).
\[
\frac{\sqrt{1+x}-1}{x}\cdot\frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}
=\frac{1}{\sqrt{1+x}+1}\xrightarrow[x\to 0]{}\frac12.
\]
3) \(0\cdot\infty\)
Rewrite the product as a fraction to convert it into \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).
\[
x\ln x = \frac{\ln x}{1/x}
\quad\text{(often becomes }\frac{-\infty}{\infty}\text{ or }\frac{0}{0}\text{ after rewriting).}
\]
4) \(\infty-\infty\)
Combine terms into a single fraction or rationalize. This removes catastrophic cancellation.
\[
\sqrt{x^2+1}-x
=\frac{(\sqrt{x^2+1}-x)(\sqrt{x^2+1}+x)}{\sqrt{x^2+1}+x}
=\frac{1}{\sqrt{x^2+1}+x}\xrightarrow[x\to\infty]{}0.
\]
5) Exponential indeterminate forms: \(1^{\infty}\), \(0^{0}\), \(\infty^{0}\)
Use logarithms. If \(y=f(x)^{g(x)}\), then
\[
\ln y = g(x)\ln f(x).
\]
Now \(\ln y\) often becomes a product like \(0\cdot\infty\), which you rewrite as a fraction and evaluate.
Finally, exponentiate: \(y=e^{\lim \ln y}\).
\[
\left(1+\frac{1}{x}\right)^x
\Rightarrow \ln y = x\ln\left(1+\frac{1}{x}\right)\xrightarrow[x\to\infty]{}1
\Rightarrow y\to e.
\]
6) Numerical confirmation
After rewriting, you can confirm by checking values close to the approach point (left/right if needed),
and by graphing. Numerical checks help detect divergence or side mismatch (DNE).
