dee9bb…: “Amid the happy solution”

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Can this limit be directly computed by limit laws? $\lim _{x\to 0} xe^{1/x}$

Can this limit be directly computed by limit laws? $\lim _{x\to 0} \frac {2x^3-1}{e^x-1}$

Can this limit be directly computed by limit laws? $\lim _{x\to 0} \frac {x+1}{5^{e^x-1}}$

Compute: $\lim _{x\to 0} \frac {x+1}{5^{e^x-1}} = \answer {1}$

Can this limit be directly computed by limit laws? $\lim _{x\to 1}{(x-1)\cdot \ln (x\ln (x))}$

Can this limit be directly computed by limit laws? $\lim _{x\to 0} x\ln x$

Can this limit be directly computed by limit laws? $\lim _{x\to 0}\frac {2^x-1}{3^{x-1}}$

Compute: $\lim _{x\to 0}\frac {2^x-1}{3^{x-1}}\begin {prompt} =\answer {0}\end {prompt}$

On the other hand the limit in the numerator is

$\begin{align*} \lim_{x\to 0}(2^x-1) &=\lim_{x\to0}(2^x)-\lim_{x\to0}(1)\\ &=1-1\\ &=0 \end{align*}$

The limits in both the numerator and denominator exist and the limit in the denominator does not equal $0$ , so we can use the Quotient Law. We find: $\frac {\lim _{x\to 0}(2^x-1)}{\lim _{x\to 0}(3^{x-1})} =\frac {0}{\frac {1}{3}}=0.$

Can this limit be directly computed by limit laws? $\lim _{x\to 0}(1+x)^{1/x}$

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