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.
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.
Of course! I'll explain how to solve each problem step by step.
1. To find the limit of the given function as x approaches negative infinity, we need to evaluate the behavior of the function as x values become more and more negative.
First, let's examine the degree of the numerator and denominator of the function. In this case, both the numerator and denominator have the highest degree term of x^3.
To find the limit as x approaches negative infinity, we need to compare the coefficients of the highest degree terms in the numerator and denominator. In this case, the coefficient of x^3 in both the numerator and denominator is 3.
So, the limit of the function as x approaches negative infinity is given by the ratio of the coefficients of the highest degree terms, which is 3/3 = 1.
Therefore, the limit of (3x^3+x^2+1)/(x^3+1) as x approaches negative infinity is equal to 1.
2. To find the slope of the tangent line to the graph of the given function (f(x) = 2x + 7) at any point, we can use the concept of derivative.
The slope of the tangent line is equal to the derivative of the function with respect to x. In this case, the function f(x) = 2x + 7 is a linear function, and the derivative of a linear function is simply the coefficient of x.
So, the derivative of f(x) = 2x + 7 is 2.
Therefore, the slope of the tangent line to the graph of the function f(x) = 2x + 7 at any point is 2.
It's important to note that when finding the slope of the tangent line to a curved function, we would use calculus techniques such as finding the derivative using the limit definition. However, in this case of a linear function, the slope is constant and equal to the coefficient of x.