Common Errors in Algebra and Calculus (Accessible HTML)

This is a compilation of common arithmetic and conceptual mistakes in most lower division math courses. These are conveyed with examples (rather than general forms) and brief explanations for each topic.

Algebra

Zero factorial does not equal zero because n factorial equals the fraction with numerator n plus one factorial, and denominator n plus one so Zero factorial equals the fraction with numerator zero plus one factorial, and denominator zero plus one, which equals one
One divided by zero does not equal zero . Observe the graph of f of x equals one divided by x at x equals zero .
x squared times x cubed does not equal x to the sixth power because x squared times x cubed equals the quantity x times x, times the quantity x times x times x times x, which equals x to the fifth power
The square root of two plus the square root of three does not equal the square root of five (Sometimes you just cannot 'simplify')
The quantity x plus y squared does not equal x squared plus y squared because The quantity x plus y squared equals the quantity x plus y times the quantity x plus y, which equals the quantity x plus y times x, plus the quantity x plus y times y, which equals x squared plus two x y plus y squared
Three times the quantity x plus y squared does not equal the quantity three x plus three y squared because Three times the quantity x plus y squared equals three times the quantity x squared plus two x y plus y squared, which equals three x squared plus six x y plus three y squared
The square root of sixteen does not equal plus or minus four because The square root of sixteen equals the square root of two to the fourth power, which equals two to the power of four halves, which equals two squared, which equals four . However, the solutions to x squared equals sixteen are x equals four or x equals negative four .
Similarly, Negative two squared does not equal four but The quantity negative two squared equals four . Parentheses are important!
The equation Two x squared equals x is not the same as Two x equals one because Two x equals one does not have the same solutions as Two x squared equals x . This 'reduction' does not simplify the equation Two x squared equals x ; it changes it.

Calculus

The limit as x approaches two of the fraction with numerator x squared minus four, and denominator x minus two, does not equal the fraction with numerator the quantity x minus two times the quantity x plus two, and denominator x minus two , but The limit as x approaches two of the fraction with numerator x squared minus four, and denominator x minus two, equals the limit as x approaches two of the fraction with numerator the quantity x minus two times the quantity x plus two, and denominator x minus two, which equals four . Cannot drop the limit between steps or else the statement does not make sense.
The derivative with respect to x of the natural log of x squared does not equal one divided by x squared because The derivative with respect to x of the natural log of x squared equals the derivative with respect to x of two times the natural log of x, which equals two divided by x
Also, The integral of one divided by x with respect to x equals the natural log of x does not imply The integral of one divided by x squared with respect to x equals the natural log of x squared
The derivative with respect to x of e to the power of x does not equal x times e to the power of x minus one because x is a variable; it is not fixed.
The derivative with respect to x of e to the power of four x does not equal e to the power of four x but rather four times e to the power of four x
The integral of the square root of x with respect to x equals two thirds times x to the power of three halves does not imply The integral of the square root of x squared plus one with respect to x equals two thirds times the quantity x squared plus one to the power of three halves
Using L'Hospital's Rule: It states, "Let lim stand for the limit The limit as x approaches c , The limit as x approaches c from the left , The limit as x approaches c from the right , The limit as x approaches infinity , or The limit as x approaches negative infinity , and suppose that The limit of f of x and The limit of g of x are both zero or are both positive infinity. If The limit of the fraction with numerator f prime of x and denominator g prime of x has a finite value or if the limit is plus or minus infinity, then The limit of the fraction with numerator f of x and denominator g of x equals the limit of the fraction with numerator f prime of x and denominator g prime of x ." This means one cannot simply state The limit of the fraction with numerator f of x and denominator g of x equals the limit of the fraction with numerator f prime of x and denominator g prime of x ; all the premises must be stated before the conclusion.
The integral of x, plus two is ambiguous; this could be interpreted as The integral of x with respect to x, plus two . The integral of the quantity x plus two with respect to x is much more clear. And remember "+C" at the end of the solution!
The integral of sine of theta with respect to x does not make sense. The variables must match!
Finally, "Euler" is pronounced "Oy-ler" not "You-ler".

Portland State University — Learning Center — Academic Integrity Reminder: Tutors may not provide answers to graded homework.

This accessible HTML version was initially generated with Claude AI by Anthropic and finalized with Gemini AI by Google.