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Evaluate: lim x→∞x^2(1-cos 1/x) Single correct option
Evaluate:
lim x → ∞ x 2 ( 1 − cos 1 x ) \displaystyle\lim_{x\to\infty}x^2\left(1-\cos\dfrac1x\right) x → ∞ lim x 2 ( 1 − cos x 1 ) A. 0 0 0 B. 1 1 1 C. ∞ \infty ∞ D. 1 2 \dfrac12 2 1 Show answer and solution Answer: (D) 1 2 \dfrac12 2 1
Put
u = 1 x → 0 u=\frac1x\to0 u = x 1 → 0 : the limit becomes
1 − cos u u 2 → 1 2 \frac{1-\cos u}{u^2}\to\frac12 u 2 1 − c o s u → 2 1 . The tempting answer
0 0 0 takes
1 − cos 1 x → 0 1-\cos\frac1x\to0 1 − cos x 1 → 0 and ignores the growing
x 2 x^2 x 2 ,
∞ \infty ∞ takes only
x 2 → ∞ x^2\to\infty x 2 → ∞ , and
1 1 1 is the limit of
sin u u \frac{\sin u}{u} u s i n u . The two factors compete, which is the
∞ ⋅ 0 \infty\cdot0 ∞ ⋅ 0 form.
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