Question

Find the limit (if it exists). $$ \lim _{\Delta x \rightarrow 0} \frac{2(x+\Delta x)-2 x}{\Delta x} $$

Short answer

The limit of the function as \( \Delta x \) approaches 0 is \( 2 \).

Step-by-step solution

Simplify the numerator

The numerator of the function is given by \( 2(x+\Delta x)-2x \). By distributing the 2 in the bracket and simplifying the equation, we get \( 2x+2\Delta x-2x = 2\Delta x \).

Rewrite the function and cancel the common factor

Our function becomes \( \frac{2\Delta x}{\Delta x} \). As \( \Delta x \) is a common factor in the numerator and denominator, we can cancel it out. So, the function is simplified to \( 2 \).

Substitute \( \Delta x = 0 \)

Now that we have simplified the function, we can substitute \( \Delta x = 0 \). Doing so has no effect on the function because all terms containing \( \Delta x \) have been canceled out. Hence, the result is \( 2 \).