Q.Evaluate $\displaystyle\lim_{x\to0}\frac{(e^{2x}-1)-\ln(1+3x)}{x}$.

Split the single fraction into two, using the difference rule from the algebra of limits, then apply the two scaled standard limits.\n> [!TLDR]\n> Split into $\frac{e^{2x}-1}{x}-\ …

Rewrite the single fraction as a difference of two fractions (valid since both share the same denominator $x$):

By the algebra of limits' difference rule (§3), the limit of this difference is the difference of the two limits, PROVIDED each piece's limit exists on its own:

Each piece is a scaled standard limit from §5: $\lim_{x\to0}\dfrac{e^{2x}-1}{x}=2$ (using $k=2$) and $\lim_{x\to0}\dfrac{\ln(1+3x)}{x}=3$ (using $k=3$).

Combining: …

Numeric cross-check (shown above): direct evaluation at a small $x=0.001$ gives approximately $-0.9935$, converging toward $-1$ as $x$ shrinks, ind …

A common error is applying the standard limits BEFORE splitting the fraction — trying to force $(e^{2x}-1)-\ln(1+3x)$ itself into one of the two named patterns directly, which does not match either formula; the fraction must …