How to Interpret and Calculate Limits of a Function

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Limits are values that say something about what happens to an expression when a variable approaches a certain value. The value of the limit can be , , or any number on the number line.

We use a specific notation for limits:

lim xaexpression

You read this as “the limit of expression as x approaches a”, or “the limit of expression as x tends to a”.

Here, “ lim” is an abbreviation for “limit”. Mathematicians try to make things as simple and intuitive as possible.

Rule

Limits of Polynomials

For the polynomial

f(x) = anxn + a n1xn1 + + a 2x2 + a 1x + a0

f(x) = anxn + a n1xn1 + + a 2x2 + a 1x + a0

the limit

lim xaf(x)

is equal to the function value f(a) for all a . Therefore,

lim xaf(x) = f(a)

Example 1

Find the limit of f(x) = x2 + 3x 4 when x 3

lim x3f(x) = lim x3x2 + 3x 4 = 32 + 3 3 4 = 9 + 9 4 = 14

Let f(x) be a polynomial of degree n. Thus

f(x) = anxn + a n1xn1 + + a 2x2 + a 1x + a0

f(x) = anxn + a n1xn1 + + a 2x2 + a 1x + a0

If x approaches plus or minus infinity, then anxn eventually becomes the dominant term in the expression, and it will determine whether the function value of f(x) is positive or negative.

Consider the limit of f(x) when x . The dominant term affects the sign of the function value of f(x), since xn tends to positive infinity if n is an even number, and it tends to negative infinity if n is an odd number. Below is a summary of the different cases:

Rule

Limits of Polynomials When x Tends to Plus/Minus Infinity

For an > 0 and n an even number, you’ve got

lim xf(x) = and lim xf(x) =

lim xf(x) = and lim xf(x) =

For an < 0 and n an even number, you’ve got

lim xf(x) = and lim xf(x) =

lim xf(x) = and lim xf(x) =

For an > 0 and n an odd number, you’ve got

lim xf(x) = and lim xf(x) =

lim xf(x) = and lim xf(x) =

For an < 0 and n an odd number, you’ve got

lim xf(x) = and lim xf(x) =

lim xf(x) = and lim xf(x) =

In any case, a polynomial tends to either plus or minus infinity when x ±.

Example 2

Find the limit of lim x 2x3 2x + 5

Here f(x) is a polynomial of degree 3 and thus of odd degree. In addition, the coefficient of the highest degree term is negative. The limit is therefore

lim x 2x3 2x + 5 =

Rule

Trick for Finding Limits of Rational Functions

  • If both the numerator and the denominator in a fraction tend to zero when x a, you can factorize the numerator and denominator separately to find the limit of the fraction lim xaf(x) g(x).

  • If both the numerator and the denominator in a fraction tend to infinity when x a, you can divide all the terms in the expression by the highest power of x in the expression.

Note! When you have a simple fraction 1 x and x , then 1 x = 0. All fractions where x is only found in the denominator approach 0 when x approaches infinity. Neat!

Rule

Some Important Limits

When a {0}, the following are true:

lim xa x = 0 lim xa x = 0 lim x0a x =

lim xa x = 0 lim xa x = 0 lim x0a x =

Note! is not a number! So no matter how big a number you choose on the number line, is infinitely larger. This means that the ratio between x and a when x or x is always infinitely large.

Example 3

Find the limit of lim x1x2 x x 1

You then have

lim x1x2 x x 1 = lim x1x(x 1) x 1 = lim x1x = 1

lim x1x2 x x 1 = lim x1x(x 1) x 1 = lim x1x = 1

Example 4

Find the limit of lim x2x2 3x x2 + 2

You then have

lim x2x2 3x x2 + 2 = lim x2x2 x2 3x x2 x2 x2 + 2 x2 = lim x2 3 x 1 + 2 x2 = 2 0 1 + 0 = 2

Example 5

Find the limit of lim x0x2 2x x

You then have

lim x0x2 2x x = lim x0x(x 2) x = lim x0x(x 2) x = lim x0x 2 = 2

Rule

Limits of Exponential Functions

For 0 < a < 1 you’ve got

lim xax = 0 = and = lim xax =

For a > 1 you’ve got

lim xax = = and = lim xax = 0

For a = 1 you’ve got

lim x1x = lim x1x = 1

Example 6

Look at the function f(x) = e0.5x. Is the function value approaching zero or infinity when x ?

Here, you must rewrite the function to find the growth factor, since it’s not obvious whether the growth factor is greater or less than 1:

e0.5x = (e0.5)x 1.648x

Thus, you see that

lim xf(x) =

Note! You could have seen that e0.5 > 1 since ex is a strictly increasing function and e0 = 1.

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