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Question: \(\lim_{x \rightarrow \infty}\frac{x^{2} + 3x + 5}{4x + 1 + x^{k}}\) exists if...

lim⁡x→∞x2+3x+54x+1+xk\lim_{x \rightarrow \infty}\frac{x^{2} + 3x + 5}{4x + 1 + x^{k}} exists if

A

k = 2

B

k < 2

C

k ≥ 2

D

None of these

Answer

k ≥ 2

Explanation

Solution

Put x = 1/y ∴ lim⁡y→0(1+3y+5y2)yk−24yk−1+yk+1\lim _ { y \rightarrow 0 } \frac { \left( 1 + 3 y + 5 y ^ { 2 } \right) y ^ { k - 2 } } { 4 y ^ { k - 1 } + y ^ { k } + 1 }

limit will give a definite value if k ≥ 2