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Calculus: Limits of Functions (Part 2 of 5)

This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.

This article will introduce limits.

There are many calculus references, the one I like to use is Calculus by Larson, Edwards and Hostetler.

This also assumes you are familiar with Python or can stumble your way through it.

Limit: the value a function approaches as it gets close to a given input coordinate.

Asymptote: the value f(x) approaches as x approaches infinity if f(x) approaches a constant value.

Convergence: a function or series is convergent if the function or series evaluates to a real number. If it doesn't it is considered divergent.

Formal Definition of a Limit
Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L iff:
limx->c-=L and limx->c+=L (i.e. the same from the left and from the right).

For functions of more than one variable the definition is the same, but extended to all variables simultaneously (e.g. an area shrinks to one point, a volume shrink to one point, etc).

Common Symptoms of Non-Existent Limits

  1. f(x) approaches a different number from the left and right
  2. f(x) increases/decreases without bound
  3. f(x) oscillates

Definition of Continuity
A function f is continuous at c if the following three conditions are met:

  1. f(c) is defined
  2. limx->c f(x) exists
  3. limx->c = f(c)

Functions can also be open on an interval or everywhere continuous (infinite).

Properties of Limits

General Form lims->c f(x)=L; lims->c g(x)=K
Constant limx->c b=b
Variable limx->c x=c
Scalar Power limx->c xn=cn
Scalar Multiple limx->c b*f(x)=b*L
Sum/Difference limx->c [f(x) +/- g(x)]=L +/- K
Product limx->c f(x)*g(x)=L*K
Quotient limx->c f(x)/g(x)=L/K; providing K != 0
Power limx->c f(x)n=Ln
Radical limx->c radn(f(x))=radn(L)

Where b, c, n, L and K are constants, x is a variable and providing the limit exists.

Limits of Vector Functions
If r is vector-valued function r(t) = f(t)i + g(t)j + h(t)k, then

limt->ar(t) = [limt->a f(t)i] + [limt->a g(t)j] + [limt->a h(t)k]

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