MCV4U1

3.3 - The Limit of a function

The limit of a function is one of the basic concepts in all of calculus. They arise when trying to find the tangent to a curve, calculating velocity, accelerationand other rates of change.The limit is the value that the dependent variable approaches when the dependentvariable approaches a specific value.

Limit Notation:

The limit of the function f(x) as x approaches a, equals the constant L.

Note: The value of a limit depends only on the functions behaviour near the value of "a", NOT AT "a".

Ex.) Find the

a) Graphicallyb) Using a Table of

Valuesc) Using Direct

Substitutiona) Graphically

Ex.) Continued....

b) Table of Values

c) Direct Substitution

f(2) = (2)2 - 1 = 3

x 1 1.5 1.9 1.99 2 2.01 2.1 2.5 3

y = x2-1

As we approach 2 from the left f(x) gets closer to _________

As we approach 2 from the right f(x) gets closer to _________

In the previous example, the Limit of the function as x approaches 2, was equal tothe value of the function, if we directly substitute x = 2 into f(x). Functions that exhibitthis property are called "Continuous" at the specified x value.

However, this is NOT always the case. Some functions are undefined at the limiting value of x.

∴ f(x) is continuous at "a"

Ex.) Find Using direct substitution we obtainwhich has NO MEANING.

In this case we need some methodof simplifying the function.

For NOW, try factoring first!!!!

Ex.) Evaluate the following limits algebraically.

a) b)c)

One-Sided LimitsSome functions that we will encounter requires us to examine the function's

behaviour on either side of the "x" value that we are approaching. These limits are known asone-sided limits.

Left-hand Limits:

Right-hand Limits:

The limit of f(x) as x approaches "a" from the LEFT.

The limit of f(x) as x approaches "a" from the RIGHT.

In general, the limit of a function at "a" only exists IF:

Ex.) Determine the following limits using one-sided limits.

a) Find Given

b) Find and Given

Homework: p. 98 - 99 # 4 - 14 (omit #12)

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