Evaluate displaystylelim_(x→0)(sin(7x)-7x)/x^3

Evaluate $d i s pl a y s t y l e l i m_{x \to 0} \frac{s in ( 7 x ) - 7 x}{x ^{3}}$

medium 3 marks Class 12 · Mathematics · Continuity and Differentiability

Answer

- \frac{343}{6}

The correct answer is $- \frac{343}{6}$ — 5. Take limit as $x \to 0$ : $- \frac{343}{6}$ .

1. Recall the Maclaurin series: $s in y = y - \frac{y ^{3}}{6} + O (y^{5})$ .
2. Substitute $y = 7 x$ : $s in (7 x) = 7 x - \frac{( 7 x ) ^{3}}{6} + O (x^{5}) = 7 x - \frac{343 x ^{3}}{6} + O (x^{5})$ .
3. Compute numerator: $s in (7 x) - 7 x = 7 x - \frac{343 x ^{3}}{6} + O (x^{5}) - 7 x = - \frac{343 x ^{3}}{6} + O (x^{5})$ .
4. Divide by $x^{3}$ : $\frac{- \frac{343 x ^{3}}{6} + O ( x ^{5} )}{x ^{3}} = - \frac{343}{6} + O (x^{2})$ .
5. Take limit as $x \to 0$ : $- \frac{343}{6}$ .

Common mistake: Forgetting the factor

7^{3}

in the numerator

Concept tested: Limit using series

Published 14 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked