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MATH 1910
Section 2.2 The Limit of a Function
Ex. What do the y-values of the graph of (^) f x x x
( ) = sin approach as the x-values approach 0? Look at a table of values for the function Look at the graph as x Æ 0 as x Æ 0 (use ASK mode) (ZOOMDEC)
Based on the numerical data and the graphical data the graph seems to approach y = 1 as x Æ 0. But, the function is not defined at x = 0. Look at the graph with the axes turned off. (2nd^ FORMAT, AxesOff)
You can barely tell, but there is a hole in the graph at x = 0. The function is not defined there.
DEFINITION OF LIMIT : We write
lim ( ) x a
f x L →
= and say “the limit of f(x) as x approaches a, equals L ”
if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to equal a.
When evaluating limits, you’re actually finding the “intended value” of the function as it approaches a certain x- value.
In the above example the function f ( ) x = (sin x ) / x is not defined at x = 0, but we can say that the limit as x approaches 0 does equal 1 because that is the “intended” y-value the graph approachs.
So lim sin x
x → (^0) x
does exist and lim sin x
x → (^0) x
= 1, regardless of whether or not the function is defined there.
MATH 1910
Section 2.2 The Limit of a Function
Ex. Evaluate these limits either graphically or numerically (using your table):
a) lim x
x x →
2
2 2 3 5 b) lim x
x x →− x
SOLUTION:
a) there’s no trouble just plugging in the value x = 2 into the function. This would give us the value of the limit as x Æ 2. You should get a limit of L = 7.
b) if you consider a table of values, you notice you can’t simply plug in x = – 1 because the function isn’t defined there, but as the table suggests, the y-values approach a limit of – 5, so
lim x
x x →− x
Left hand and Right hand limits : When using your table to evaluate these limits you’ve considered values to the left and to the right of the central x value you’re approaching. These are called the left hand and right hand limits of the function f (x).
lim ( ) x a
f x → −^
= L (taking values of x approaching a from the left, x < a)
lim ( ) x a
f x → +^
= L (taking values of x approaching a from the right, x > a)
IMPORTANT! Æ The limit lim ( ) x a
f x L →
= exists ONLY if lim ( ) x a
f x → −^
= L AND lim ( ) x a
f x → +^
= L
Ex. Given the following graph evaluate the following quantities:
a) lim ( ) x
f x → 2 −
b) lim ( ) x
f x → 2 + c) lim ( ) x
f x → 2
d) f ( 2 )
e) (^) x lim→ 7 − f ( ) x f) (^) x lim→ 7 + f ( ) x g) lim x → 7 f ( ) x h) f ( 7 )
Ex. Sketch a graph of an example of a function f ( x ) that satisfies the following conditions:
x lim→− 1 − f^ ( ) x^ =^4 x lim→− 1 + f^ ( ) x^ = −^2 f^ ( 1)−^ is undefined lim x → 3 f ( ) x = 0 f (3) = 6
