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Magazine Problem 4
Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=\cosh x ; a=0 ; n=4 $$
Problem 5
Evaluate the limit, if it exists. $$ \lim _{x \rightarrow 0} \frac{x}{\tan x} $$
Problem 5
Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it. $$ \int_{0}^{+\infty} x e^{-x} d x $$
Problem 5
Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=\ln x ; a=1 ; n=3 $$
Problem 6
Evaluate the limit, if it exists. $$ \lim _{x \rightarrow 0} \frac{\tan x-x}{x-\sin x} $$
Problem 6
Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=\sqrt{x} ; a=4 ; n=4 $$
Problem 6
Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it. $$ \int_{3}^{+\infty} \frac{d x}{\sqrt{x-1}} $$
Problem 7
Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=\ln \cos x ; a=\frac{1}{3} \pi ; n=3 $$
Problem 7
Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it. $$ \int_{-x}^{+x} x \cosh x d x $$
Problem 7
Evaluate the limit, if it exists. $$ \lim _{x \rightarrow+\infty} \frac{\sin \frac{2}{x}}{\frac{1}{x}} $$
