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 SolutionsQuestion 1True or False. If a function f is not defined at x = a then the limitlim 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 2True or False. If f and g are two functions such thatand 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 3True 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 4True 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 5If 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 6True 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 7True 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 8True 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 9True 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 .
