Limits of Exponential and Logarithmic Functions

The chapter states, without re-proving, eight standard exponential and logarithmic limits (listed in the table note above), the two most-used being $\lim_{x\to0}\dfrac{a^x-1}{x}=\log a$ (any base $a>0$) and $\lim_{x\to0}(1+x)^{1/x}=e$, together with the logarithmic counterpart $\lim_{x\to0}\dfrac{\log(1+x)}{x}=1$. Nearly every exercise in this section is one of three recurring shapes.\n\nShape 1 — a bare exponential ratio. Dividing numerator and denominator by x turns each piece into the standard $(a^x-1)/x\to\log a$ form; e.g. $\lim_{x\to0}\dfrac{5^x-1}{\sin x}=\dfrac{\lim(5^x-1)/x}{\lim(\sin x)/x}=\dfrac{\log5}{1}=\log5$, and a difference of two such ratios simply subtracts, e.g. $\lim_{x\to0}\dfrac{5^x-3^x}{x}=\log5-\log3=\log(5/3)$.\n\nShape 2 — a power tending to $e^{(\cdot)}$. Whenever the limit has the form $[1+f(x)]^{1/x}$ with $f(x)\to0$, rewrite the bracket's exponent to match $f(x)$: $[1+f(x)]^{1/x}=\left\{[1+f(x)]^{1/f(x)}\right\}^{f(x)/x}$, so the inner brace tends to e and the whole limit becomes $e^{\lim f(x)/x}$. E.g. for $\lim_{x\to0}\left[1+\dfrac{5x}{6}\right]^{1/x}$, here $f(x)=5x/6$ and $f(x)/x=5/6$, giving $e^{5/6}$; for $\lim_{x\to0}\left[\dfrac{3x+2}{2-5x}\right]^{1/3x}$, first write $\dfrac{3x+2}{2-5x}-1=\dfrac{8x}{2-5x}$ so $f(x)/(3x)\to\dfrac{8}{2\times3}=\dfrac43$, giving $e^{4/3}$.\n\nShape 3 — a genuine log-difference, handled by dividing into the $\log(1+kx)/x\to k$ form. E.g. $\lim_{x\to0}\dfrac{\log4+\log(0.25+x)}{x}$ first combines the two logs into $\log[4(0.25+x)]=\log(1+4x)$, then the standard result gives $4\times1=4$.\n\nMixed forms combining exponential AND log AND trig pieces are solved by peeling each standard limit off in turn and multiplying/dividing the results — e.g. $\lim_{x\to0}\dfrac{e^{2x}+e^{-2x}-2}{x\sin x}$ is engineered into $\left[\dfrac{e^{2x}-1}{2x}\right]^2\times\dfrac{4}{e^{2x}}\times\dfrac{x}{\sin x}\to1^2\times4\times1=4$ by first multiplying the whole fraction by $e^{2x}/e^{2x} …