maths

• Rational expression


  A fraction, usually with variables in it
• Simplifying rational expressions to lowest terms

  1. Factor the numerator

  2. Factor the denominator

  3. Cancel common factors
• Multiplying rational expressions


  1. Multiply the numerators

  2. Multiply the denominators
• Dividing rational expressions


  Multiply by the reciprocal of the denominator fraction
• Adding and subtracting rational expressions

  1. Find the least common denominator

  2. Rewrite each fraction in terms of the LCD

  3. Add or subtract the numerators
• Secant line


  A line that stretches between two points on the graph of a function
• Average rate of change

  The slope of the secant line between two points on the graph of a function
• Difference quotient


  The average rate of change using the notation x and x+h
• Limit


  As x approaches a, f(x) approaches L, even if f(a) ≠ L
• The limit doesn't care about the value of f(a), it only cares about the values of f(x) as x approaches a
• Limit


  As x approaches a, f(x) approaches L, meaning f(x) gets arbitrarily close to L as x gets arbitrarily close to a
• The limit as x approaches a of f(x) does not care about the value of f at x=a, it only cares about the values of f when x is near a
• If the limit as x approaches a from the left is not equal to the limit as x approaches a from the right

  The limit as x approaches a does not exist
• If there is a vertical asymptote at x=a

  The limit as x approaches a does not exist
• If the function exhibits "wild behavior" near x=a

  The limit as x approaches a does not exist
• One-sided limit


  The limit as x approaches a from the left or the right side
• The limit as x approaches a from the left is denoted with a superscript minus sign: lim(x→a-)
• The limit as x approaches a from the right is denoted with a superscript plus sign: lim(x→a+)
• If the one-sided limits from the left and right exist but are not equal, the overall limit does not exist
• If the one-sided limits from the left and right both exist and are equal, the overall limit exists and is equal to that value
• If a function has a vertical asymptote at x=a, the one-sided limits as x approaches a from the left and right do not exist as finite numbers, but may exist as positive or negative infinity
• If a function exhibits "wild behavior" near x=a, the limit as x approaches a does not exist
• The limit laws only apply if the limits of the component functions exist as finite numbers
• Squeeze Theorem


  If f(x) ≤ g(x) ≤ h(x) for x near a, and lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L
• The product rule only applies when the component limits both exist. Since the second limit doesn't exist, the product rule tells us nothing about whether the limit that we're interested in exists or doesn't
• We can use the squeeze theorem to evaluate the limit
• Squeeze theorem


  • Allows evaluating limits when a function is trapped between two other functions with the same limit
• In the example, x^2 sin(1/x) is trapped between x^2 and -x^2, which both have a limit of 0 as x goes to 0
• Therefore, the limit of x^2 sin(1/x) as x goes to 0 is also 0
• Limits of the form 0/0 are called indeterminate forms
• Steps to evaluate 0/0 limits

  Simplify the expression

  2. Use algebraic tricks like factoring, multiplying by a conjugate, etc.

  3. Apply limit laws
• The limit law for quotients states the limit of f(x)/g(x) is the quotient of the limits, provided the limit of g(x) is not 0
• If the limit of the denominator is 0, then the limit may still exist if the numerator also goes to 0
• If both the numerator and denominator go to 0, the limit may not exist
• As x approaches 3 from the left

  The limit is positive infinity
• As x approaches 3 from the right


  The limit is negative infinity
• Since the limits from the left and right are different, the overall limit does not exist
• As x approaches -4 from the left


  The limit is negative infinity
• As x approaches -4 from the right

  The limit is positive infinity
• The limit as x goes to negative four of 5x over the absolute value of x plus four is equal to negative infinity