Evaluating Limits Analytically (College Board AP® Calculus AB)

If a function is continuous on an open interval, and if is a number contained in that interval, then

• 

• I.e. for a continuous function, the limit at a point is equal to the value of the function at that point

This idea can be extended to a point that is on or 'just outside' the border of an interval on which a function (or part of a function) is defined

• This is especially useful for one-sided limits at the 'joins' of piecewise functions

• E.g. the function defined by

  • Note that and are both continuous functions on all the real numbers

  • is on the border for the piece of , and 'just outside' the border for the piece of

  • So by substituting,

    • and

Sometimes trying to evaluate a limit by substitution gives an answer that is not defined

• E.g. returning a value like

In these cases, sometimes the equation of the function can be simplified

• For example by factorising and cancelling common factors

• After this substitution may give a well-defined value for the limit

E.g. let be the function defined by

• This function is undefined at , because

• But to find the limit at we don't care about the function's value at

  • We only care about it's behavior near

• So we can cancel the common factor and proceed like this:

  • 

    • The 'vertical bar notation' means to evaluate the expression to the left of the bar at

• Note that the limit can be well-defined even at a point where the function itself is undefined!

• Also note that and are not the same function

  • The function is defined at , and the function isn't

  • But for purposes of calculating limits, canceling common factors to make 'look like' is perfectly okay

Not all limits for which substitution gives can be resolved this way

• See the L'Hospital's Rule study guide for another method to deal with such limits

A quotient involving surds (i.e. square roots) can sometimes be rewritten by multiplying it top and bottom by a conjugate of either the numerator or denominator

• The conjugate of is

  • and the conjugate of is

For example,

This can be used to find limits where just using substitution gives an undefined expression like

A quotient involving powers of x can be rewritten by multiplying its top and bottom by the reciprocal of the highest power of x in the numerator or denominator

For example,

This can be used to find a limit at infinity

• because if is any constant and is a positive integer, then

  •