Difficult Limit Problems and Solutions

The list of tougher limit questions in calculus to learn how to use the limit rules while finding the limits of difficult functions and solutions for hard limits problems with understandable steps in different methods.

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x-\sin{x}}{x^3}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \Big(\dfrac{\sin{x}}{x}\Big)^{\dfrac{1}{x^2}}}$

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \bigg(\dfrac{1}{x^2}-\dfrac{1}{\sin^2{x}}\bigg)}$

$\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\cos{(\sin{x})}-\cos{x}}{x^4}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{x^x-x^a}{a^x-a^a}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{x+\log_{e}{\sqrt{x^2+1}-x}}{x^3}}$

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.