Limits

Calculus Level 4

lim ⁡ x → 0 x 2 cos ⁡ ( 5 x ) − tan ⁡ 2 ( 3 x ) − 1 \large \lim_{x\to 0} \frac{x^{2}}{\cos(5x)-\tan^{2}(3x)-1} Evaluate the limit above without using L'Hôpital rule.

-1.14E-2 -4.65E-2 3.14E-2 -2.65E-2

Majed Musleh by Majed Musleh , 28, Palestine

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1 solution

cos ⁡ 5 x \cos 5x ~ 1 − ( 5 x ) 2 2 ! 1 - \frac{(5x)^2}{2!} when x → \rightarrow 0.

tan ⁡ 2 ( 3 x ) \tan^{2} (3x) ~ ( 3 x ) 2 (3x)^{2} when x → \rightarrow 0. lim ⁡ x → 0 x 2 cos ⁡ ( 5 x ) − t a n 2 ( 3 x ) − 1 = lim ⁡ x → 0 x 2 1 − 25 x 2 2 − 9 x 2 − 1 = \lim_{x \to 0} \frac{x^2}{\cos (5x) - tan^{2}(3x) - 1} = \lim_{x \to 0} \frac{x^2}{1 - \frac{25x^2}{2} - 9x^2 - 1} = = lim ⁡ x → 0 x 2 x 2 ( − 25 2 − 9 ) = 2 − 43 = − 4.65 E − 2 = \lim_{x \to 0} \frac{x^2}{x^2(\frac{-25}{2} - 9)} = \frac {2}{-43}= \boxed{-4.65E-2}

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