Let f be a differentiable function. Prove that if \(\lim_{x\to\infty} (f(x)+f'(x))=1\), then \(\lim_{x\to\infty} f(x)=1\).
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Let f be a differentiable function. Prove that if \(\lim_{x\to\infty} (f(x)+f'(x))=1\), then \(\lim_{x\to\infty} f(x)=1\).
Evaluate the following limit:
\(\displaystyle \lim_{\varepsilon\to 0}\int_0^{2\varepsilon} \log\left(\frac{|\sin t-\varepsilon|}{\sin \varepsilon}\right) \frac{dt}{\sin t}\).
Let \(a_1<\cdots\) be a sequence of positive integers such that \(\log a_1, \log a_2,\log a_3,\cdots\) are linearly independent over the rational field \(\mathbb Q\). Prove that \(\lim_{k\to \infty} a_k/k=\infty\).
Let \(a_1=\sqrt{1+2}\),
\(a_2=\sqrt{1+2\sqrt{1+3}}\),
\(a_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4}}}\), …,
\(a_n=\sqrt{1+2\sqrt{1+3\sqrt {\cdots \sqrt{\sqrt{\sqrt{\cdots\sqrt{1+n\sqrt{1+(n+1)}}}}}}}}\), … .
Prove that \(\displaystyle\lim_{n\to \infty} \frac{a_{n+1}-a_{n}}{a_n-a_{n-1}}=\frac12\).