November 2020

Find the limit of $ \lim\limits_{n \to  \infty } \sqrt[n]{n} $.  

Calculate the upper and lower Riemann Sum for the function $f\left(x\right)=1$ when $x$ is rational and $f\left(x\right)=-1$ when $x$ is irrational. Hence... Read More

Calculate the limit \lim\limits_{x \to 0 } x \ln x without using L’Hôpital’s rule.

Does the following converge or diverge? $\int_{0}^{1} \ln x dx$

Does the following converge or diverge? $\int_{0}^{1} x^{-1} dx$

Does the following converge or diverge? $\int_{0}^{1} x^{-\frac{1}{2}} dx$

With the help of Rolle’s Theorem and the function $f\left(x\right)=e^{-x} \left(x-a\right)\left(x-b\right)$ prove that the equation $\left(x-a\right)\left(x-b\right)=\left(x-a\right)+\left(x-b\right)$ will have a solution in the... Read More

Does the following converge or diverge? $\int_{0}^{\infty} e^{-x} dx$

Prove that polynomials with real coefficients and odd degree always have at least one real root.

Find the Maclaurin Series of $\frac{4x^2-3}{\left(1-x\right)\left(1-2x\right)^2}$ where $x$ is small i.e. less than $\frac{1}{2}$ in magnitude. But, you can’t use the formula... Read More