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in Limits and Derivatives by (15.2k points)
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Evaluate the following:

(i) \(\lim \limits_{x \to 0} \cfrac{e^{ax}-1}{e^{bx}-1}\)

lim (e^ax - 1) / (e^bx - 1) [x ∈ 0]

(ii) \(\lim \limits_{x \to 0} \cfrac{log(1+x)}{sin x}\)

lim (log(1+x)) / sinx [x ∈ 0]

(iii) \(\lim \limits_{x \to 0} \cfrac{e^{x}+e^{-x}-2}{x^2}\)

lim (e^x + e^-x -2) / x^2 [x ∈ 0]

(iv) \(\lim \limits_{x \to 0} \cfrac{e^{x}- sinx -1}{x}\)

lim (e^x + sinx -1) / x [x ∈ 0]

(v) \(\lim \limits_{x \to 0} \cfrac{e^{sin x}- 1}{log(1+x)}\)

lim (e^sinx -1) / log(1+ x) [x ∈ 0]

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1 Answer

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by (15.7k points)

(i) \(\lim \limits_{x \to 0} \cfrac{e^{ax}-1}{e^{bx}-1}\)

(ii) \(\lim \limits_{x \to 0} \cfrac{log(1+x)}{sin x}\)

\(\lim \limits_{x \to 0} \cfrac{\frac{log(1+x)}{x}}{\frac{sin x}{x}}\) = \(\frac{1}{1}\) = 1

(iii) 

(iv) 

(v)

\(\frac{1}{1} \times 1 = 1.\)

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