Proof Prove that if the limit of f ( x ) as x approaches c exists, then the limit must be unique. [ Hint : Let lim x → c f ( x ) = L 1 and lim x → c f ( x ) = L 2 and prove that L 1 = L 2 . ]
Proof Prove that if the limit of f ( x ) as x approaches c exists, then the limit must be unique. [ Hint : Let lim x → c f ( x ) = L 1 and lim x → c f ( x ) = L 2 and prove that L 1 = L 2 . ]
Solution Summary: The author explains that if f(x) as x approaches c exists, then the limit must be unique.
Proof Prove that if the limit of f (x) as x approaches c exists, then the limit must be unique. [Hint: Let
lim
x
→
c
f
(
x
)
=
L
1
and
lim
x
→
c
f
(
x
)
=
L
2
and prove that
L
1
=
L
2
.
]
Definition of infinite limit: Let X⊆ R, f: X -> R and a∈ X'. If for every M>0 there exists delta > 0 such that |f(x)| > M whenever x∈X and 0< |x-a| < delta then we say that the limit as x approaches a of f(x) is ∞ which is denoted as lim {x-> a} f(x) = ∞.
Suppose a∈R, ∈>0, and f,g : N*(a,∈) ->R. If lim {x-> a} f(x) = L>0 and lim {x-> a} g(x)= ∞, prove lim {x-> a} (fg)(x)=∞.
(Right and Left Limits). Introductory calculus coursestypically refer to the right-hand limit of a function as the limit obtained by“letting x approach a from the right-hand side.”
(a) Give a proper definition in the style of Definition 4.2.1 ((Functional Limit).for the right-hand and left-hand limit statements:
limx→a+f(x) = L and limx→a−f(x) = M.
(b) Prove that limx→a f(x) = L if and only if both the right and left-handlimits equal L.
Computing a limit graphically and analytically
Chapter 2 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
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