Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 106: 30

$$28$$

Given $$\lim_{x\to 0} \frac{\tan3x^2+\sin^25x}{x^2} $$ Then \begin{align*} \lim_{x\to 0} \frac{\tan3x^2+\sin^25x}{x^2}&=\lim_{x\to 0} \frac{\tan3x^2 }{x^2}+\lim_{x\to 0} \frac{ \sin^25x}{x^2}\\ &=\lim_{3x^2\to 0} \frac{3\tan3x^2 }{3x^2}+25\left(\lim_{x\to 0} \frac{ \sin 5x}{5x }\right)^2\\ &=3(1)+25(1)\\ &=28 \end{align*}