Estimate the limit numerically. $$\lim _ { x \rightarrow - \infty } \frac { x ^ { 6 } + 7,000 x ^ { 3 } + 800,000 } { 2 x ^ { 6 } + 1,269 x ^ { 3 } }$$
A) $\frac { 1 } { 2 }$
B) $- \infty$
C) 0
D) $+ \infty$
E) $- \frac { 1 } { 2 }$

Question

Estimate the limit numerically.   $$\lim _ { x \rightarrow - \infty } \frac { x ^ { 6 } + 7,000 x ^ { 3 } + 800,000 } { 2 x ^ { 6 } + 1,269 x ^ { 3 } }$$  
A) $\frac { 1 } { 2 }$
B) $- \infty$
C) 0
D) $+ \infty$
E) $- \frac { 1 } { 2 }$

Quizplus · Accepted Answer

As $x$ approaches $-\infty$, the terms with the highest degree in both the numerator and denominator dominate the behavior of the function. Here, those are $x^6$ in the numerator and $2x^6$ in the denominator. The coefficients of these terms are 1 and 2, respectively, so their ratio is $\frac{1}{2}$, which is the value the function approaches.

Estimate the Limit Numerically lim⁡x→−∞x6+7,000x3+800,0002x6+1,269x3\lim _ { x \rightarrow - \infty } \frac { x ^ { 6 } + 7,000 x ^ { 3 } + 800,000 } { 2 x ^ { 6 } + 1,269 x ^ { 3 } }x→−∞lim​2x6+1,269x3x6+7,000x3+800,000​

Estimate the Limit Numerically $\lim _ { x \rightarrow - \infty } \frac { x ^ { 6 } + 7,000 x ^ { 3 } + 800,000 } { 2 x ^ { 6 } + 1,269 x ^ { 3 } }$