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Question: Evaluate\(\lim _ { x \rightarrow - \infty }\) \(\frac{x^{4}\sin\frac{1}{x} + x^{2}}{1 + |x|^{3}}\)...

Evaluatelim⁡x→−∞\lim _ { x \rightarrow - \infty } x4sin⁡1x+x21+∣x∣3\frac{x^{4}\sin\frac{1}{x} + x^{2}}{1 + |x|^{3}}

A

–1

B

1

C

0

D

None of these

Answer

–1

Explanation

Solution

Let x = –y. Then y → ∞ when x → –∞ 

∴ limit = lim⁡y→∞\lim _ { y \rightarrow \infty } y4sin⁡−1y+y21+∣−y∣3\frac { y ^ { 4 } \sin \frac { - 1 } { y } + y ^ { 2 } } { 1 + | - y | ^ { 3 } } = lim⁡y→∞\lim _ { y \rightarrow \infty } −y4sin⁡1y+y21+y3\frac { - y ^ { 4 } \sin \frac { 1 } { y } + y ^ { 2 } } { 1 + y ^ { 3 } }

[Q y is positive]

= lim⁡y→∞\lim _ { y \rightarrow \infty }

= lim⁡y→∞\lim _ { y \rightarrow \infty }

= 1 . −11\frac { - 1 } { 1 } + 0 = –1.