Answer:
The end behaviour of this function is -ꝏ when x approaches +ꝏ and +ꝏ when x approaches -ꝏ. The leading term is -3x^5.
Step-by-step explanation:
The end behaviour can be found by taking the limits of function for x approaching +ꝏ and -ꝏ:
lim x→+/-ꝏ (-3x^5+9x^4+5x^3+3)
To solve the limits it must be factorized the highest term in the polynomial:
lim x→+/-ꝏ {-3x^5*[-3x^5/(-3x^5)+9x^4/(-3x^5)+5x^3/(-3x^5)+3/(-3x^5)]}
By simplifying:
lim x→+/-ꝏ {-3x^5*[1-3/x-5/3x^2-1/x^5)]}
If we first take the limits in the within the square brackets, we will notice than whether the limits approaches +ꝏ or -ꝏ the limits of each of these terms tend to zero due to the denominator is greater than the numerator:
lim x→+/-ꝏ {-3x^5*[1-0-0-0)]}
lim x→+/-ꝏ {-3x^5*[1]}
lim x→+/-ꝏ {-3x^5}
This way we find the leading term. Now taking the positive and negative limits:
lim x→+ꝏ {-3x^5} = -ꝏ
and:
lim x→-ꝏ {-3x^5} = +ꝏ
