find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using…

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it.\n lim_{x\rightarrow0^{+}}left(\frac{1}{x}-\frac{1}{e^{x}-1}\right)

find the limit. use lhospitals rule where appropriate. if there is a more elementary method, consider using it.\n lim_{x\rightarrow0^{+}}left(\frac{1}{x}-\frac{1}{e^{x}-1}\right)

Answer

Explanation:

Step1: Combine fractions

$\lim_{x\rightarrow0^{+}}\frac{e^{x}-1 - x}{x(e^{x}-1)}$

Step2: Check form and apply L'Hopital's

It's $\frac{0}{0}$ form. Differentiate: $\lim_{x\rightarrow0^{+}}\frac{e^{x}-1}{e^{x}-1+xe^{x}}$

Step3: Apply L'Hopital's again

Still $\frac{0}{0}$ form. Differentiate: $\lim_{x\rightarrow0^{+}}\frac{e^{x}}{e^{x}+e^{x}+xe^{x}}$

Answer:

$\frac{1}{2}$