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

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