Question

find the indicated limits, if they exist.

1. lim (3x^3+x^2+1)/(x^3+1)
x-> -infinity
2. find the slope of the tangent line to the graph of the given function at any point:

f(x) = 2x +7

Could anyone please, please, please explain to me how to do these problems? It seems like sometimes I can do the problem and I think I know it but then I get another one wrong. I need to know what I'm messing up on.

Answer 1

For a function composed of a rational function, the limit to infinity can be determined by the quotient of the leading terms of the numerator and denominator:
Lim 3x^3/x^3 x->∞
= Lim 3 x->∞
=3

The slope of the tangent to the (line) is the coefficient of x, namely 2.
In a general way, the tangent function is obtained by taking the derivative of the function, i.e. d(2x+7)/dx = 2.