The Master Formula Box
Standard Trig Limit: $\lim_{x \to 0} \frac{\sin x}{x} = 1$Standard Log Limit: $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$Exponential Limit: $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
Dimensional Analysis of Limits
A limit $\lim_{x \to c} f(x)$ must have the same dimensions as $f(x)$. In Physics, if you are finding the limit of a force function as time approaches zero, your result must be in Newtons. If your mathematical manipulation leads to a result with different dimensions, you have likely made an error in the derivation of the function itself.
Variations: Limits at Infinity
When $x \to \infty$ for a rational function $P(x)/Q(x)$:
- If degree of $P <$ degree of $Q$, the limit is $0$.
- If degrees are equal, the limit is the ratio of the leading coefficients.
- If degree of $P >$ degree of $Q$, the limit is $\pm \infty$.
Shortcuts & Mnemonics
- Sandwich Theorem: If $g(x) \leq f(x) \leq h(x)$ and both $g$ and $h$ approach $L$, then $f$ must also approach $L$. Think of it as "Squeezing" the value.
- Mnemonic: "Hopital the Top-ital and the Bot-ital." (Reminding you to differentiate top and bottom separately).
- Small Angle: For limits as $x \to 0$, $\sin x$ and $\tan x$ behave exactly like $x$.
Edge Cases
- Oscillating Limits: $\lim_{x \to 0} \sin(1/x)$ does not exist because the function bounces between $-1$ and $1$ infinitely fast as it nears zero.
- L'Hopital Failure: L'Hopital's rule only works if the limit is indeterminate. Using it on a defined limit (like $1/2$) will give a wrong answer.
- Continuity vs. Differentiability: Every differentiable function is continuous, but not every continuous function is differentiable (e.g., $y = |x|$).
