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Grade 11Differential Calculus

If lim(x->0) [1+x + (f(x)/x)]1/x = e3
Find f(x). f(0) not equal to zero.

Profile image of Kawal
12 Years agoGrade 11
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1 Answer

Profile image of Jitender Singh
ApprovedApproved Tutor Answer12 Years ago
Ans:Hello student, please find answer to your question
\lim_{x\rightarrow 0}(1+x+\frac{f(x)}{x})^{\frac{1}{x}} = e^{3}
\lim_{x\rightarrow 0}(1+\frac{x^{2}+f(x)}{x})^{\frac{1}{x}} = e^{3}
\lim_{x\rightarrow 0}(1+\frac{x^{2}+f(x)}{x})^{\frac{x^{2}+f(x)}{x}.\frac{x}{x^{2}+f(x)}\frac{1}{x}} = e^{3}
\lim_{x\rightarrow 0}(1+\frac{x^{2}+f(x)}{x})^\frac{x}{x^{2}+f(x)} = e
\lim_{x\rightarrow 0}1+\frac{f(x)}{x^{2}} = 3\lim_{x\rightarrow 0}(1+\frac{f(x)}{x^{2}}) = 3
\lim_{x\rightarrow 0}(\frac{f(x)}{x^{2}}) = 2
f(x) = kx^{2}
k = 2
f(x) = 2x^{2}

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