# Questions and Answers on Limits in Calculus

A set of questions on the concepts of the limit of a function in
calculus are presented along with their answers. These questions have been designed to help you **gain deep understanding of the concept of limits ** which is of major importance in understanding calculus concepts such as the
derivative and integrals of a function. These questions also helps you find out concepts that need reviewing.

## Questions with Solutions

### Question 1

**True or False**. If a function f is not defined at x = a then the limit

lim f(x) as x approaches a

never exists.

__Answer :__False. lim f(x) as x approaches a may exist even if function f is undefined at x = a. The concept of limits has to do with the behaviour 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

and

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.

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 thatand

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 .