Consider the polynomial function $$ f(t)=t^{8}+4 t^{4}-t^{2}-2 $$ As $$ t \rightarrow-\infty, \quad f(t) \rightarrow $$ ? As $$ t \rightarrow \infty, \quad f(t) \rightarrow $$ ?

Question

UpStudy AI · Accepted Answer

1. The given polynomial is 
   $$
   f(t) = t^8 + 4t^4 - t^2 - 2.
   $$
   
2. The dominant term of any polynomial is the term with the highest degree. Here, the dominant term is 
   $$
   t^8.
   $$
   
3. Since the degree of the dominant term is 8 (which is even) and its coefficient is positive, the behavior of $$ f(t) $$ as $$ t \rightarrow \pm\infty $$ is determined by $$ t^8 $$.

4. When $$ t \rightarrow \infty $$:
   $$
   t^8 \rightarrow \infty,
   $$
   implying 
   $$
   f(t) \rightarrow \infty.
   $$

5. When $$ t \rightarrow -\infty $$:
   $$
   (-t)^8 = t^8 \rightarrow \infty,
   $$
   which also implies 
   $$
   f(t) \rightarrow \infty.
   $$

Therefore, as $$ t \rightarrow -\infty $$, $$ f(t) \rightarrow \infty $$; and as $$ t \rightarrow \infty $$, $$ f(t) \rightarrow \infty $$.
As $$ t $$ approaches both positive and negative infinity, $$ f(t) $$ approaches infinity.

Consider the polynomial function \( f(t)=t^{8}+4 t^{4}-t^{2}-2 \) As \( t \rightarrow-\infty, \quad f(t) \rightarrow \) ? As \( t \rightarrow \infty, \quad f(t) \rightarrow \) ?

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions