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Limits

Subject:

Calculus [1]

Subject X2:

Calculus [1]

Limits

The concept of limits is essential to calculus. The limit is the basis for all calculus problems. A good understanding of limits will help explain many theories in calculus.

Consider a function *f* defined for values of x, as x gets close to a number *a*, not necessarily true for x = *a*. If the value of *f*(x) approaches a number b as x approaches *a*, then the limit of *f*(x) as x approaches *a* is equal to* b*, denoted as :

lim_{x}_{®}_{a}*f*(x) = *b*

ex.

Find the limit of *f*(x) = 3x + 2 as x approaches 3.

It is said that as x approaches 3, 3x approaches 9, and 3x + 2 approaches 11. Thus;

lim_{x}_{®}_{3} 3x + 2 = 11

Find the limit of *f*(x) = 1/ x - 3 as x approaches 3.

It is said that as x approaches 3, x - 3 approaches 0, and 1/ x - 3 approaches 1/ 0 which is undefined. Thus;

lim_{x}_{®}_{3} 1/x - 3 = undefined

In limits, the only thing that matters is how a function is defined near the point *a*.

Subject:

Calculus [1]

Subject X2:

Calculus [1]

Elementary Limits:

lim_{x}_{®}_{a}*f*(x) = *f*(x)

This limit represents a horizontal line, which says that as x approaches *a*,

*f*(x) = *f*(x).

lim_{x}_{®}_{a}*f*(x) = a

This limit says that as x approaches *a*,* f*(x) also approaches a.

Subject:

Calculus [1]

Subject X2:

Calculus [1]

Addition, Subtraction, Multiplication and Division of Limits:

If lim_{x}_{Â®}_{a}*f*(x) = *S* and lim_{x}_{Â®}_{a}*g*(x) = *T* then,

lim_{x}_{Â®}_{a} [*f*(x) - *g*(x)] = *S + T*

lim_{x}_{Â®}_{a} [*f*(x) - *g*(x)] = *S - T*

lim_{x}_{Â®}_{a} [*f*(x) - *g*(x)] = *S Â· T*

Subject:

Calculus [1]

Subject X2:

Calculus [1]

Limits Of Polynomials and Rational Functions:

If P(x) and Q(x) are polynomials, then:

lim_{x}_{Â®}_{a} P(x) = P(a)

and

Limits can be approached from the negative ( or left ) or the positive ( or right ) side of a number denoted as:

lim_{x}_{Â®}_{a}_{-}*f*(x) = b* *or lim_{x}_{Â®}_{a}_{+}*f*(x) = b

If lim_{x}_{Â®}_{a}_{-}*f*(x) Â¹ lim_{x}_{Â®}_{a}_{+}*f*(x) , then

lim_{x}_{Â®}_{a}*f*(x) does not exist

If lim_{x}_{Â®}_{a}_{-}*f*(x) = lim_{x}_{Â®}_{a}_{+}*f*(x) = b , then

lim_{x}_{Â®}_{a}*f*(x) = b

Subject:

Calculus [1]

Subject X2:

Calculus [1]

Infinite Limits:

If the value of *f*(x) gets larger and larger without bound as x approaches *a*, then :

lim_{x}_{®}_{a}_{-}*f*(x) = +

Similarly; If the value of *f*(x) gets smaller and smaller without bound as x approaches *a*, then :

lim_{x}_{®}_{a}*f*(x) = -¥

Subject:

Calculus [1]

Subject X2:

Calculus [1]

Limits at Infinity:

Consider a function *f* defined for large positive ( or negative ) values of x, as x increases indefinitely in the positive ( or negative ) direction. If the value of *f*(x) approaches a number *b* as x increases (or decreases ) indefinitely , then the **limit** of *f*(x) as x increases (or decreases ) indefinitely is equal to *b*, denoted as :

lim_{x}_{®}_{+}_{¥} *f*(x) = *b* or lim_{x}_{®}_{-}_{¥} *f*(x) = *b*

Subject:

Calculus [1]

Subject X2:

Calculus [1]

Continuity:

A function *f* is continuos at x = a if *f* is defined at x = a and either ; *f* is not defined anywhere near *a*; or *f* is defined arbitrarily near x = a and lim_{x}_{®}_{a}*f*(x)=*f*(a).

Conversely, A function *f* is discontinuous at x = a if *f* is defined at x = a and *f* is not continuos at x = a.

Addition, Subtraction, Multiplication and Division of Continuity:

If *f* and *g* are continuos functions at x = a then; *f + g*, *f - g, f · g* and *f/g*; where *g*(a) ¹ 0; are also continuous at x = a.

If the function *f* is a polynomial or a rational function then *f* is continuos wherever it is defined.

Given *f*(*g*(x)), where *g* is continuos at x = a , and *f* is continuos at x = *g*(a) then *f*(*g*(x)) is continuos at x = a.

Subject:

Calculus [1]

Subject X2:

Calculus [1]

More Graphing

Intercepts and Asymptotes:

Given an equation of the form:

*f*(x) = P(x) / Q(x) in reduced form;

If (x - a)^{n} (where n is a positive integer) is a factor of either P(x) or Q(x) and if (x - a)^{n+1} is a factor of neither , then

a. The graph crosses the x axis at x = a if and only if n is odd.

b. The graph stays on the same side of the x axis at x = a if and only if n is even.

Vertical Asymptotes:

The line x = a is a vertical asymptote of the function *f* if at least one of the following statements are true:

lim_{x}_{®}_{a}_{+}f(x) = +¥ |
lim_{x}_{®}_{a}_{+}f(x) = -¥ |

lim_{x}_{®}_{a}_{-}f(x) = +¥ |
lim_{x}_{®}_{a}_{-}f(x) = -¥ |

Horizontal Asymptotes:

The line y = b is a horizontal asymptote of the function *f* if at least one of the statements is true:

lim_{x}_{®}_{+}_{¥} *f*(x) = *b* or lim_{x}_{®}_{-}_{¥} *f*(x) = *b*

Slant Asymptotes:

The line y = ax + b is a slant asymptote of a function *f* if:

lim_{x}_{®}_{±}_{ ¥} [ *f*(x) - (ax + b)] = 0

Subject:

Calculus [1]

Subject X2:

Calculus [1]

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