# 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 help 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

\( \lim_{x \to a} f(x) \)

never exists.

__Answer :__False.

\( \lim_{x \to a} f(x) \) 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 \[ \lim_{x \to a} f(x) = +\infty \] and \[ \lim_{x \to a} g(x) = +\infty \] then \( \lim_{x \to a} [ f(x) - g(x) ] \) is always equal to 0.

__Answer :__False.

Infinity is not a number and \( \infty - \infty \) is not equal to 0. \( +\infty \) is a symbol to represent large but undefined numbers. \( -\infty \) is a symbol to represent small but undefined numbers.

### 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.

\( f(x) = \dfrac{x - 2}{(x - 1)(x + 3)} \)

The degree of the denominator (2) is higher than the degree of the numerator (1) hence the graph of \( f \) 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) \( \lim_{x \to a} [f(x) \cdot g(x)] \) is always equal to 0

(B) \( \lim_{x \to a} [f(x) \cdot g(x)] \) is never equal to 0

(C) \( \lim_{x \to a} [f(x) \cdot g(x)] \) may be \( +\infty \) or \( -\infty \)

(D) \( \lim_{x \to a} [f(x) \cdot g(x)] \) may be equal to a finite value.

__Answer :__(C) and (D).

Try the following functions:

\( f(x) = \dfrac{1}{x} \) and \( g(x) = 2x \) as \( x \) approaches 0.

\( f(x) = \dfrac{1}{x^2} \) and \( g(x) = x \) as \( x \) approaches 0.

### Question 6

**True or False**. If \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) exist as \( x \) approaches \( a \) then \( \lim_{x \to a} \left[\dfrac{f(x)}{g(x)}\right] = \dfrac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \) as \( x \) approaches \( a \).

__Answer :__False.

Only if \( \lim_{x \to a} g(x) \) is not equal to 0.

### Question 7

**True or False**. For any polynomial function \( p(x) \), \( \lim_{x \to a} p(x) \) is always equal to \( p(a) \).

__Answer :__True.

All polynomial functions are continuous functions and therefore \( \lim_{x \to a} p(x) = p(a) \).

### Question 8

**True or False**. If \( \lim_{x \to a^-} f(x) = L_1 \) as \( x \) approaches \( a \) from the

__left__and \( \lim_{x \to a^+} f(x) = L_2 \) as \( x \) approaches \( a \) from the

__right__, \( \lim_{x \to a} f(x) \) exists only if \( L_1 = L_2 \).

__Answer :__True.

This is an important property of the limits.

### Question 9

**True or False**. \( \lim_{x \to \infty} \sin x \) as \( x \) approaches very large values (\(+\infty\)) is \( +1 \) or \( -1 \).

__Answer :__False.

\( \sin x \) is an oscillating function and has no limit as \( x \) becomes very large (\(+\infty\)) or very small (\(-\infty\)). The same can be said about \( \cos x \).