Math 151: 2.6 Limits at Infinity
WARM-UP
a) Does log x 2 = 2log x for all values of x? Explain.
b) Evaluate
9 and explain the value using the graph of a function.
c) Sketch the graph of y = x 2 .
()
1
by thinking graphically for
x→∞ f x
Evaluate lim f x and lim
x→∞
1
d) f x =
x
( )
( )
x
e) f x = 2
x +1
()
x
f) f x = 2
x −1
( )
x3
g) f x = x
3
()
( )
h) f x =
ln x
x
END BEHAVIOUR: The end behaviour is the zoomed out behaviour of a function. The left
and right end behaviours of f are represented by limits as x → −∞ or x → ∞ :
( )
Left End Behaviour: lim f x
x→−∞
()
Right End Behaviour: lim f x
x→∞
Exercise 1: Determine the left and right end behaviours for
()
lim f ( x ) =
f x =
a) x 2
b) −x 3 + 4x − 1
c) tan−1 x
d) e x
e) ln x
x→−∞
( )
lim f x =
x→∞
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Math 151: 2.6 Limits at Infinity
END BEHAVIOUR MODEL
An end behaviour model of a function is a simpler (monomial) function that has the same
zoomed out behaviour (at its “ends”, as x → ±∞ or x → ∞ ) as the original function.
( )
f x =
2x 2 + 3x − 1
1
= 2x + 3 − (blue) and g ( x ) = 2x + 3 (purple) and h x = 2x (red)
x
x
()
x ∈ ⎡⎣ −7,7 ⎤⎦
x ∈ ⎡⎣ −30,30 ⎤⎦
x ∈ ⎡⎣ −600,600 ⎤⎦
()
When zoomed out far enough, the original f x looks like its end behaviour model 2x .
•
When x is small (relatively close to zero), the lower order terms of a function are very
important to the value of the overall function.
•
However, as x gets larges in magnitude, the lower-order terms become negligible, and
the leading-order term dominates in size/numerical value.
Two functions f and g have the same
()
()
•
left end behaviour model if lim f x = lim g x ⇔ lim
•
right end behaviour model if lim f x = lim g x ⇔ lim
x→−∞
x→∞
()
x→−∞
x→∞
()
x→−∞
x→−∞
=1
=1
Exercise 2: For each function, state the end behaviour models. Confirm using limits.
()
( )
a) f x = sin 2x
()
b) f x = x 2 − x + e − x
1
c) f x = x +
x
()
© Raelene Gibson 2017
()
d) f x = 3x − 5 +
()
e) f x =
( )
f) f x =
2x − 1
x2 − 9
x3 − 1
x +1
4x 2 + 1
3x
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Math 151: 2.6 Limits at Infinity
( )
(
Exercise 3: Consider the polynomial p x = − 21 x + 2
) ( x − 1) ( x
2
3
2
)(
)
+1 x − 3 .
a) What is the degree of p? ____
b) Make/use a sign chart to sketch the graph of p as accurately as possible without
calculus (max/min), and state the end behaviours of p x : lim p x =
lim p x =
( )
x→−∞
( )
x→∞
()
()
p ( x)
p ( x)
lim
= 1 and lim
= 1.
m( x)
m( x)
c) State the end behaviour model(s) m x for the polynomial. Confirm your choice by
showing that
x→−∞
x→∞
End Behaviour (or Non-Vertical) Asymptotes
The
()
end
f x =
behaviour
( ) = q ( x) + r ( x)
d ( x)
d ( x)
n x
(or
•
•
non-vertical)
asymptote
for
a
rational
function
with no common factors in the numerator and denominator, and
where the degree of r x is less than the degree of d x
()
()
()
is given by y = q x .
()
( )
Exercise 4: The degree of r x < the degree of d x . Show that:
( )
()
a) q x is an end behaviour model (not necessarily a monomial) of f x .
()
()
b) f x intersects its non-VA if r x = 0 has real solutions.
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Math 151: 2.6 Limits at Infinity
Fill in the blanks:
c) y = q x is a horizontal asymptote on the x – axis if deg n x
d)
e)
f)
()
y = q ( x ) is
y = q ( x ) is
y = q ( x ) is
{ ( )} ___ deg{d ( x )} .
a HA not on the x – axis if deg {n ( x )} ___ deg {d ( x )} .
a slant asymptote if deg {n ( x )} __________________ deg {d ( x )} .
a parabolic asymptote if deg {n ( x )} _______________ deg {d ( x )} .
Exercise 5: Determine (i) the end behaviour asymptote (equation and type) for each
function and (ii) any point of intersection between f and its EBA. Graph each function.
3x − 9
4 − 3x 3
a) f x =
d) f x = 2
12 − 4x
2x + x
( )
( )
()
b) f x =
4 − 3x
2x 2 + x
( )
e) f x =
4 − 3x 2
c) f x = 2
2x + x
2x 2 − 3x − 5
4x − 10
()
( )
f) f x = −x 2 + 3 −
x +1
x2 + 1
Limit Definitions of Horizontal and Vertical Asymptotes
•
Note: a horizontal asymptote is a special type of non-vertical (end behaviour)
asymptote while a vertical asymptote is an infinite discontinuity of the function.
•
The line y = L is a ________________ asymptote of the graph of the function f if
( )
( )
either lim f x = _____ or lim f x = _____ , or both.
x→___
•
x→___
The line x = a is a ________________ asymptote of the graph of the function f if:
( )
( )
( )
( )
lim f x = ___ , lim f x = ___ , lim f x = ___ , lim f x = ___ .
x→___
x→___
© Raelene Gibson 2017
x→___
x→___
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Math 151: 2.6 Limits at Infinity
Exercise 6: Use limits to determine the equations and types of all asymptotes for
2x 2 − 3
a) f x =
1− x − 3x 2
1− x + 3x 3
b) f x =
x2 − 2
()
()
( )
c) f x =
x
4x 2 + 1
The Squeeze Theorem also applies for limits of functions as x → ±∞ , as long as the
function is bounded by two other functions whose limits as x → ±∞ are equal, say L.
Then, the limit as x → ±∞ of the given function will also be L.
cos x
cos x
and lim
using the Squeeze Theorem and confirm
x→∞
x→−∞
x
x
Exercise 7: Evaluate lim
graphically.
Using Properties to Evaluate Limits
The Properties of Limits that are valid as x approaches a finite value are still valid as
x → ±∞ (see Constant, Sum, Difference, Product, Constant Multiple, Quotient, and Power
Rules for limits in 2.2).
2x − cos x
Exercise 8: Evaluate lim
using the properties of limits and confirm graphically.
x→∞
3x
Using the Reciprocal Transformation Substitution: It may be useful to use the substitution
t = 1x to change the one-sided limit as x → −∞ to 1x = t → 0 − or as x → ∞ to 1x = t → 0 + .
cos 1x
.
x→∞ 1+ 1
x
Exercise 9: Evaluate lim
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Math 151: 2.6 Limits at Infinity
Using Symmetry to Evaluate Limits: The following limit property holds
•
•
( )
for odd functions: If lim f ( x ) = L
( ) ()
and f ( −x ) = −f ( x ) , then
( )
lim f ( x ) = −L .
for even functions: If lim f x = L and f −x = f x , then lim f x = L .
x→c
x→c
x→−c
x→−c
Exercise 10: Evaluate lim sin x cos x if lim sin x cos x = M , for a ∈° and a > 0 .
x→−a
Exercise 11: Evaluate lim
x→∞
x→a
x
4x 2 + 1
( )
expressions to the graph of f x =
and lim
x→−∞
x
4x 2 + 1
4x 2 + 1
, and state the meaning of the limit
.
Hint: Consider symmetry and the identity for
© Raelene Gibson 2017
x
x 2 in your answer.
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