DERIVATIVES USING THE LIMIT DEFINITION

The following problems require the use of the limit definition of a derivative, which is given by

.

They range in difficulty from easy to somewhat challenging. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by making proper use of functional notation and careful use of basic algebra. Keep in mind that the goal (in most cases) of these types of problems is to be able to divide out the term so that the indeterminant form of the expression can be circumvented and the limit can be calculated.

• PROBLEM 1 : Use the limit definition to compute the derivative, f'(x), for

  .

  Click HERE to see a detailed solution to problem 1.
  
  
  
  
  

• PROBLEM 2 : Use the limit definition to compute the derivative, f'(x), for

  .

  Click HERE to see a detailed solution to problem 2.
  
  
  
  
  

• PROBLEM 3 : Use the limit definition to compute the derivative, f'(x), for

  .

  Click HERE to see a detailed solution to problem 3.
  
  
  
  
  

• PROBLEM 4 : Use the limit definition to compute the derivative, f'(x), for

  .

  Click HERE to see a detailed solution to problem 4.
  
  
  
  
  

• PROBLEM 5 : Use the limit definition to compute the derivative, f'(x), for

  .

  This problem may be more difficult than it first appears.

  Click HERE to see a detailed solution to problem 5.
  
  
  
  
  

• PROBLEM 6 : Use the limit definition to compute the derivative, f'(x), for

  .

  Click HERE to see a detailed solution to problem 6.
  
  
  
  
  

• PROBLEM 7 : Use the limit definition to compute the derivative, f'(x), for

  $f(x) = \displaystyle { x - 1 \over x^2 + 3x }$ .

  Click HERE to see a detailed solution to problem 7.
  
  
  
  
  

• PROBLEM 8 : Use the limit definition to compute the derivative, f'(x), for

  $f(x) = \sqrt{ x^3 - x }$ .

  Click HERE to see a detailed solution to problem 8.
  
  
  
  
  

• PROBLEM 9 : Assume that

  $f(x) = \cases{ 2 + \sqrt{ x }, & if $\space x \ge 1 $\space \cr \displaystyle{ 1 \over 2 } x + \displaystyle{ 5 \over 2 } , & if $ x < 1 $\space . \cr }$

  Show that f is differentiable at x=1, i.e., use the limit definition of the derivative to compute f'(1) .

  Click HERE to see a detailed solution to problem 9.
  
  
  
  
  

• PROBLEM 10 : Assume that

  $f(x) = \cases{ x^2 \sin \Big( \displaystyle{ 1 \over x } \Big), & if $\space x \ne 0 $\space \cr \ \ \ \ \ 0 \ \ \ \ \ , & if $ x = 0 $\space . \cr }$

  Show that f is differentiable at x=0, i.e., use the limit definition of the derivative to compute f'(0) .

  Click HERE to see a detailed solution to problem 10.
  
  
  
  
  

• PROBLEM 11 : Use the limit definition to compute the derivative, f'(x), for

  f(x) = | x² - 3x | .

  Click HERE to see a detailed solution to problem 11.
  
  
  
  
  

• PROBLEM 12 : Assume that

  $f(x) = \cases{ \displaystyle{ 1\over 4 }x^3 - \displaystyle{1 \over 2 } x^2, &... ...$\space \cr \displaystyle{ -6x-6 \over x^2+2 } , & if $ x < 2 $\space . \cr }$

  Determine if f is differentiable at x=2, i.e., determine if f'(2) exists.

  Click HERE to see a detailed solution to problem 12.
  
  
  
  
  

Click HERE to return to the original list of various types of calculus problems.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

kouba@math.ucdavis.edu

• About this document ...