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 that
\( \lim_{x \to a} f(x) = +\infty \)

and
\( \lim_{x \to a} g(x) = 0 \)

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 \( \cos x \).

Links and References

Calculus questions with answers and Calculus tutorials and problems .

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