Question 1:
True or False. If a function f is not defined at x = a then the limit
lim f(x) as x approches a
never exists.
Answer :
False. lim f(x) as x approches a may exist even if function f is indefined at x = a. The concept of limits has to do with the behavior of the function close to x = a and not at x = a.
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 approches a from the left and lim f(x) = L2 as x approches a from the right. lim f(x) as x approches 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 approches very large values (+infinity) is + 1 or  1.
Answer :
False. sin x is an oscilating 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
questions with answers and tutorials and problems .
