Evaluating a Limit Consider the limit lim x → 0 + ( − x ln x ) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. Use a graphing utility to verify the result. M FOR FURTHER INFORMATION for a geometric approach to this exercise, see the article "A Geometric Proof of lim d → 0 + ( − d ln d ) = 0 " by John H. Mathews in The College Mathematics Journal, To view this article, go to MathArticles.com.
Evaluating a Limit Consider the limit
lim
x
→
0
+
(
−
x
ln
x
)
(a) Describe the type of indeterminate form that is obtained by direct substitution.
(b) Evaluate the limit. Use a graphing utility to verify the result.
M FOR FURTHER INFORMATION for a geometric approach to this exercise, see the article "A Geometric Proof of
lim
d
→
0
+
(
−
d
ln
d
)
=
0
"
by John H. Mathews in The CollegeMathematics Journal, To view this article, go to MathArticles.com.
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.
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