__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
questions with answers and tutorials and problems .