The value of $\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{n^{2}}$ is

$\lim \limits_{x}$ ${\rightarrow \frac{1}{\sqrt{2}}} \frac{\sin \left(\cos ^{-1} x\right)-x}{1-\tan \left(\cos ^{-1} x\right)}$ is equal to

The value of $\mathop {\lim }\limits_{n \to \infty } {\left( {e \cdot {a^2} \cdot {e^3} \cdot {a^4} \cdots {e^{n - 1}} \cdot {a^n}} \right)^{\frac{1}{{{n^2} + 1}}}}$ is equal to

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 0} \frac{ax+b}{cx+1}$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $\lim _{x \rightarrow \infty} f(x)=M > 0$. Then which of the following is false?

$\mathop {\lim }\limits_{x \to \infty } \frac{{{{(2x + 1)}^{40}}{{(4x - 1)}^5}}}{{{{(2x + 3)}^{45}}}} = $