Question 2:
True or False. If f and g are two functions such that
lim f(x) as x --> a = + infinity
and
lim g(x) as x --> a = + infinity
then lim [ f(x) - g(x) ] as x --> a is always equal to 0.
Answer :
False. Infinity is not a number and infinity - infinity is not equal to 0. +Infinity is a symbol to represent large but undefined numbers. -infinity is small but undefined number.
Question 3:
True or False. The graph of a rational function may cross its vertical asymptote.
Answer :
False. Vertical asymptotes are defined at x values that make the denominator of the rational function equal to 0 and therefore the function is undefined at these values.
Question 4:
True or False. The graph of a function may cross its horizontal asymptote.
Answer :
True. Here is an example.
f(x) = (x - 2) / [ (x - 1) (x + 3) ]
The degree of the denominator (2) is higher than the degree of the numerator (1) hence the graph of has a horizontal asymptote y = 0 which is the x axis. But the graph of f has an x intercept at x = 2, which means it cuts the x axis which is the horizontal asymptote at x = 2.
Question 5:
If f(x) and g(x) are such that
lim f(x) as x --> a = + infinity
and
lim g(x) as x --> a = 0
then
(A) lim [ f(x) . g(x) ] as x --> a is always equal to 0
(B) lim [ f(x) . g(x) ] as x --> a is never equal to 0
(C) lim [ f(x) . g(x) ] as x --> a may be +infinity or -infinity
(D) lim [ f(x) . g(x) ] as x --> a may be equal to a finite value.
Answer :
(C) and (D). Try the following functions:
f(x) = 1 / x and g(x) = 2x as x approaches 0.
f(x) = 1 / x^{ 2} and g(x) = x as x approaches 0.
Question 6:
True or False. If lim f(x) and lim g(x) exist as x approaches a then lim [ f(x) / g(x) ] = lim f(x) / lim g(x) as x approaches a.
Answer :
False. Only if lim g(x) is not equal to 0.
Question 7:
True or False. For any polynomial function p(x), lim p(x) as x approaches a is always equal to p(a).
Answer :
True. All polynomial functions are continuous functions and therefore lim p(x) as x approaches a = p(a).
Question 8:
True or False. If lim f(x) = L1 as x approaches a from the left and lim f(x) = L2 as x approaches a from the right. lim f(x) as x approaches a exists only if L1 = L2.
Answer :
True. This is an important property of the limits.
Question 9:
True or False. lim sin x as x approaches very large values (+infinity) is + 1 or - 1.
Answer :
False. sin x is an oscillating function and has no limit as x becomes very large (+infinity) or very small (-infinity). The same can be said about cos x.
More references on calculus
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