Formula sheet: Limits and Derivatives

1.1 Limits

As x approaches a, the limit of f(x) describes the value the function settles near, whether x comes from below or from above, and whether or not the function is defined at a itself. The left limit uses x approaching a from below and the right limit uses x approaching from above. The limit exists exactly when both one-sided limits exist and are equal, and then the common value is the limit.

Limit laws allow sums, differences, products, quotients and constant multiples to be handled term by term, provided each individual limit exists and no denominator vanishes. Polynomials and rational functions are evaluated by direct substitution whenever the denominator is non-zero; when substitution gives zero over zero, factor or rationalise before substituting again.

1.2 Derivatives

The derivative of f at a is the limit of the difference quotient as h tends to zero: the limit of (f(a plus h) minus f(a)) by h. It measures the instantaneous rate of change of the function, and geometrically it is the slope of the tangent at the point. The function is differentiable at a only when this limit exists, and differentiability implies continuity but not the other way round.

For sums, differences, constant multiples and products, the derivative follows the algebra rules: the derivative of a sum is the sum of the derivatives, the derivative of a constant times a function is the constant times the derivative, and products follow the product rule. The derivative of x to the n is n times x to the n minus 1 for any real n where the expression is defined.

1.3 Standard derivatives

The two trigonometric limits are the reason the sine derivative comes out as cosine. Dividing numerator and denominator by x and using the limit of sine x by x equal to 1, together with the limit of (1 minus cos x) by x equal to 0, gives the difference quotient of sine as a cosine in the limit. The same style of argument gives every standard derivative, so a first-principles question is never out of reach.