Questions: lim as h approaches 0 of ((x+h)/(x+h)-5) - (x/(x-5))/h

Solution Steps

To solve this limit problem, we need to simplify the expression inside the limit. The expression is a difference quotient, which suggests that it might be related to the derivative of a function. We will first simplify the numerator by finding a common denominator and then simplify the entire expression. After simplification, we can directly evaluate the limit as \( h \) approaches 0.

The given limit is: \[ \lim _{h \rightarrow 0} \frac{\frac{(x+h)}{(x+h)-5}-\frac{x}{x-5}}{h} \] First, simplify the numerator by finding a common denominator: \[ \frac{(x+h)}{(x+h)-5} - \frac{x}{x-5} = \frac{(x+h)(x-5) - x(x+h-5)}{((x+h)-5)(x-5)} \] Simplify the expression: \[ = \frac{x^2 - 5x + hx - 5h - x^2 - hx + 5x}{((x+h)-5)(x-5)} \] \[ = \frac{-5h}{((x+h)-5)(x-5)} \]

Substitute the simplified numerator back into the original limit expression: \[ \lim _{h \rightarrow 0} \frac{-5h}{h((x+h)-5)(x-5)} \] Cancel \( h \) in the numerator and denominator: \[ \lim _{h \rightarrow 0} \frac{-5}{((x+h)-5)(x-5)} \]

Evaluate the limit as \( h \) approaches 0: \[ = \frac{-5}{(x-5)^2} \]

Final Answer