# 1350 Limits

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Math 1350 - Limits and continuity

MAPLE has a powerful capability to find a limit. The MAPLE function that is used to find a limit is **limit**.

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*Example1:*

find the limit of f(x) = (x-2)/(x^2-1)

First define the function f(x)

> f:=x->(x-2)/(x^2 - 1);

Find the limit of f(x) as x approaches 0, 1, -1, infinity

> limit(f(x), x=0);

> limit(f(x), x=1);

> limit(f(x), x=-1);

> limit(f(x), x=infinity);

When x approaches 1, -1 the limit does not exist therefore MAPLE gives undefined.

To convince ourselves about the function´s behavior we can plot it.

> plot(f(x), x=-10..10, y=-10..10);

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*Example2:*

a different behavior has the limit of f(x) = (x-2)/(x-1)^2.

> f:=x->(x-2)/(x - 1)^2;

> limit(f(x), x=0);

> limit(f(x), x=1);

> limit(f(x), x=-1);

> limit(f(x), x=infinity);

> plot(f(x), x=-10..10, y=-10..10);

*Example3:*

find the limit of f(x)=|x| as x -> 0. To write in MAPLE the absolute value of x, use the command **abs**.

> f:=x->abs(x);

> limit(f(x), x=0);

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*Example3:*

define f(x) = sin(1/x). Find the limit as x -> 0, and as x -> infinity. Plot the function.

> f:=x->sin(1/x);

> limit(f(x), x=0);

> limit(f(x), x=infinity);

> plot(f(x), x=-10..10, y=-1.5..1.5);

f(x) is not continuous at as x -> 0 because its limit is ?1..1. It is continuous everywhere else since limit f(x) = 0 as x -> .

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**Exercises:**

Find the limit of the fallowing functions a) as x -> 0; b) as x->. Plot them.

- f(x) = (e^x 1)/x
- f(x) = (e^x 1)/e^x
- f(x) = 5-2*x
- f(x) = ln(x)
- f(x) = cos(x)
- f(x) = (1/x)*cos(x)
- f(x) = (1/x)*sin(1/x)