Recognising a limit as a derivative practice problems
Spot the derivative hiding in a limit: identify f and a, then evaluate f′(a). 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
The limit \( \displaystyle \lim_{h\to 0} \frac{\ln{\left(h + e \right)} - 1}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{1 - \cos{\left(h \right)}}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{\ln{\left(h + 2 \right)} - \ln{\left(2 \right)}}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{\cos{\left(h \right)} - 1}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{\sqrt{h + 16} - 4}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{\left(h - 1\right)^{4} - 1}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{\left(h + 2\right)^{4} - 16}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{\cos{\left(h + \frac{\pi}{4} \right)} - \frac{\sqrt{2}}{2}}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{- \frac{1}{2} + \frac{1}{h + 2}}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
The limit \( \displaystyle \lim_{h\to 0} \frac{\sin{\left(h \right)}}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.