Calculate the following limits using some known derivatives.

\(\displaystyle \lim_{x\to0}\frac{\cos^3x+e^x-2}{x} \qquad \lim_{x\to0}\frac{\sh x}{\log_2(1+x)} \)

Difficulty: 2.

\(\displaystyle \lim_{x\to0} {\cos(x^2)-1 \over x}=?\)

Difficulty: 3.

\(\displaystyle \lim_{x\to0} {\cos (xe^x)-\cos (xe^{-x}) \over x^3}=?\)

Difficulty: 3.

Calculate the following limits using L'Hospital's rule!

 (1)  \(\displaystyle \lim_{x\to{\pi/2}}\frac{\cos x}{\frac{\pi}2 - x}\),      (2)  \(\displaystyle \lim_{x\to 0+}x^{\sqrt x}\).

Difficulty: 3.

Calculate the following limits using L'Hospital's rule and also using the Taylor polynomial!

 (1)  \(\displaystyle \lim_{x\to 0} \frac{\sin x - x}{x^3}\),      (2)  \(\displaystyle \lim_{x\to0} \frac{\cos(x^2)-1}{x}\),      (3)  \(\displaystyle \lim_{x\to0} \frac{\cos (xe^x)-\cos (xe^{-x})}{x^3}\),      (4)  \(\displaystyle \lim_{x=\infty} \frac{1+\sqrt{x}+\sqrt[3]{x}}{1+\sqrt[3]{x}+\sqrt[4]{x}}\),      (5)  \(\displaystyle \lim_{x\to0} \frac{(1+x)^5-(1+5x)}{x^2+x^5}\),      (6)  \(\displaystyle \lim_{x\to0} \frac{\cos x - e^{-\frac{x^2}2} }{x^4}\),      (7)  \(\displaystyle \lim_{x\to0} \frac{e^x\sin x - x(1+x)}{x^3}\).

Difficulty: 3.

\(\displaystyle \lim_{x\to0}\frac{\sin 3x}{\tg 5x}=? \quad \lim_{x\to0}\frac{\log\cos ax}{\log\ch bx}=? \quad \lim_{x\to0}\left(\frac{\sin x}x\right)^{x^{-2}}=? \quad \lim_{x\to1}\left((x-1)\tg\frac{\pi x}2\right)=? \quad \lim_{x\to\infty}\frac{\sin\ln x}{x} = ? \)

Can we use the L'Hospital rule? Can we use the definition of the derivative at \(\displaystyle 0\) (or \(\displaystyle 1\))?

Difficulty: 3.

\(\displaystyle \lim_{x\to0}\frac{2e^x+e^{-x}-3}{\sin 2x+x^2+\sh x}=? \qquad \lim_{x\to1}x^{\frac1{1-x}}=? \qquad \lim_{x\to1}(2-x)^{\tg\frac{\pi x}2}=? \qquad \lim_{x\to\infty}\frac{2x+\sin x}{2x-\cos x} =? \)

Can we use the L'Hospital rule? Can we use the definition of the derivative at \(\displaystyle 0\) (or \(\displaystyle 1\))?

Difficulty: 3.

Can we use the L'Hospital rule for \(\displaystyle \frac{0}{\text{anything}}\) type limits?

Difficulty: 4.

Assume that \(\displaystyle f,g\) are \(\displaystyle k\) times differentiable, \(\displaystyle \lim\limits_\infty|g|=\infty\), \(\displaystyle g^{(k)}\ne0\) and \(\displaystyle \lim\limits_\infty\frac{f^{(k)}}{g^{(k)}}=\beta\). Does it imply that \(\displaystyle \lim\limits_\infty\frac{f}{g}=\beta\)?

Difficulty: 4.

\(\displaystyle \lim_{x\to0} \log_{(1-x^2)} (\cos bx) = ? \qquad \lim_{x\to0}\left(\frac{1+e^x}{1+\cos x}\right)^{\ctg x}=? \qquad \lim_{x\to0}\frac{2\cth (x^2)-\ctg(1-\cos x)}{\ln(1+x)-\sin x}=? \)

Difficulty: 4.

\(\displaystyle \lim_{x\to1} (x-1)^{\log_x2}=? \qquad \lim_{x\to0}\left(\ch x\right)^{\ctg^2 x} =? \)

Difficulty: 4.

\(\displaystyle \lim_{x\to0}\frac{\displaystyle\ctg x-\frac1x}{3^x-\ch x} = ? \)

Difficulty: 5.

\(\displaystyle \lim_{x\to0} \left(\frac1{\sin x}-\frac1{e^x-1}\right) = ? \)

Difficulty: 5.

\(\displaystyle \lim_{x\to0}\dfrac{\cth x-\ctg x}{\ln(1+x)-x}=? \)

Difficulty: 5.