Question

If $\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$exists, it equals the slope of the ______ _____ to the graph of f at the point $\left(c,f\left(c\right)\right).$

Short answer

If $\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$exists, it equals the slope of thetangentlineto the graph offat the point$\left(c,f\left(c\right)\right).$

Step-by-step solution

Step 1. Given Information

The slope of the tangent line through a point on the graph of a function provides the function's instantaneous rate of change at that point.

Step 2. Filling the blank

As we know the tangent of a line to the graph of the function $f\left(x\right)$at the point $\left(c,f\left(c\right)\right)$has a slope of $\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}.$

So, if $\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$exists, it equals the slope of the tangent line to the graph offat the point$\left(c,f\left(c\right)\right).$