Question

Find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\). $$ f(x)=2 x+3 $$

Short answer

The limit \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} = 2\).

Step-by-step solution

Substitute the function into the limit

Replace \(f(x)\) and \(f(x + \Delta x)\) in the limit with the specific function \(2x + 3\). So, the function being evaluated under the limit becomes: \[\frac{(2(x + \Delta x) + 3) - (2x + 3)}{\Delta x}\]

Simplify the function under the limit

Expand the terms in the numerator, and then simplify by combining like terms. \[\frac{2x + 2\Delta x + 3 - 2x -3}{\Delta x} = \frac{2\Delta x}{\Delta x}\]

Further simplify the function

Now we cancel out \(\Delta x\) in the numerator with \(\Delta x\)\ in the denominator to get:\( 2\)

Apply the limit

As \(\Delta x\) approaches 0, the expression becomes the constant \(2\). So, the final expression under the limit is \(2\).