Question

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=2 x+4 ;(1,6) $$

Short answer

The slope of the tangent line to the graph of the function \(f(x)=2x+4\) at the point (1,6) is 2.

Step-by-step solution

Understand the Problem

Firstly, grasp the problem which is to find the slope of the tangent line at a given point for the function \(f(x)=2x+4\). The point is at (1,6). So we will use the limit definition of the derivative.

Start with the formula

Now, apply the limit definition of a derivative: \(f′(a)=lim_{h→0} [f(a+h)−f(a)]/h\), replacing a with 1 in our case. This yields: \(f′(1)=lim_{h→0}[f(1+h)−f(1)]/h\)

Apply Function To Limit

Next, apply the value to your function. Put \(f(1+h)\) and \(f(1)\) in place of x in your function \(f(x) = 2x + 4\) and simplify: \(f′(1)=lim_{h→0}[(2(1+h)+4-2(1)-4)/h] = lim_{h→0}[2h/h]\)

Cancel Out The Variables

Simplify the limit equation by cancelling out the variable h in the numerator with the variable h in the denominator. So it will be: \(f′(1)=lim_{h→0}[2]=2\)

Interpret the Result

So the slope of the tangent line going through the point (1,6) is 2. This slope is consistent with the slope of the overall function, which is also 2, indicating that the function is a straight line. A straight line has the same slope at every point along the line, so we can confirm that our result is correct.