Learn – Calculus – Limits

Motivation and intuition

Exercises

Problem set

  1. Consider the function f(x) = \frac{x^2 - 1}{x-1}. Make a function table to find the values of f(x) at x = 1.1, 1.01, 1.001, 0.9, 0.99, 0.999. Look for a pattern in the values of f(x) as x gets closer and closer to 1. Based on the pattern, determine the value of \lim_{x\to 1}f(x).
  2. Consider the function g(a) = \frac{a^2 + a - 2}{a-1}. Make a function table to find the values of g(a) at a = 1.1, 1.01, 1.001, 0.9, 0.99, 0.999. Look for a pattern in the values of g(a) as a gets closer and closer to 1. Based on the pattern you observe, determine the value of \lim_{a\to 1}g(a).
  3. Consider the function h(t) = 3t^3-4t^2+2t-4. Make a function table to find the values of h(t) at t = 2.1, 2.01, 2.001, 1.9, 1.99, 1.999. Look for a pattern in the values of h(t) as t gets closer and closer to 2. Based on the pattern you observe, determine the value of \lim_{t\to 2}h(t).
  4. Consider the function s(h) = \frac{1-(1-h)^2}{h}. Determine the values of s(h) at h = 0.01, 0.001, 0.0001, -0.01, -0.001,-0.0001. Look for a pattern in the values of s(h) as h gets closer and closer to 0. Based on the pattern you observe, determine the value of \lim_{h\to 0}s(h).
  5. Consider the function f(x) = \frac{5x^2-16x+12}{5x-10}. Determine the values of f(x) at x = 1.9, 1.99, 1.999, 2.1, 2.01, 2.001. Look for a pattern in the values of f(x) as x gets closer and closer to 2. Based on the pattern you observe, determine the value of \lim_{x\to 2}f(x).
  6. Consider the function f(x) = \frac{e^x-1}{x}. Determine the values of f(x) at x = -0.1, -0.01, -0.001, 0.1, 0.01, 0.001. Look for a pattern in the values of f(x) as x gets closer and closer to 0. Based on the pattern you observe, determine the value of \lim_{x\to 0}f(x).
  7. Consider the function f(x) = \frac{\sin 10x}{x}, where x takes values in radians. Determine the values of f(x) at x = -0.1, -0.01, -0.001, 0.1, 0.01, 0.001. Look for a pattern in the values of f(x) as x gets closer and closer to 0. Based on the pattern you observe, determine the value of \lim_{x\to 0}f(x).
  8. Consider the function f(x) = \frac{10x}{\tan 5x}, where x takes values in radians. Determine the values of f(x) at x = -0.1, -0.01, -0.001, 0.1, 0.01, 0.001. Look for a pattern in the values of f(x) as x gets closer and closer to 0. Based on the pattern you observe, determine the value of \lim_{x\to 0}f(x).

Problem set

  1. For the function shown in the graph, evaluate the limits.

    1. \lim\limits_{x\to -2} f(x)

    2. \lim\limits_{x\to 0} f(x)

    3. \lim\limits_{x\to 1} f(x)
  2. For the function shown in the graph, evaluate the limits.

    1. \lim\limits_{x\to 0} f(x)

    2. \lim\limits_{x\to 2} f(x)

    3. \lim\limits_{x\to 4} f(x)
  3. For the function shown in the graph, evaluate the limits.

    1. \lim\limits_{x\to -1} f(x)

    2. \lim\limits_{x\to 3} f(x)

    3. \lim\limits_{x\to 4} f(x)
  4. For the function shown in the graph, evaluate the limits.

    1. \lim\limits_{x\to -1^-} f(x)

    2. \lim\limits_{x\to -1^+} f(x)

    3. \lim\limits_{x\to -1} f(x)

    4. \lim\limits_{x\to 2^-} f(x)

    5. \lim\limits_{x\to 2^+} f(x)

    6. \lim\limits_{x\to 2} f(x)

Problem set

  1. Consider the polynomial 6x^4+13x^3-90x^2-20x+30. We consider a sequence of five secants to this polynomial. Note that a secant intersects the graph of the polynomial at two points. For all the secants that we consider, one of the points of intersection has x-coordinate equal to 2. The secants differ from each other in their second points of intersection. The x-coordinates for the second points of intersection of the five secants are respectively as follows: 1.9, 1.99, 1.999, 1.9999, 1.99999.

    1. Find the slope of each of the five secants.
    2. Can you now guess the slope of the tangent to the polynomial at x = 2?
  2. When a function is drawn on the coordinate plane, a relevant question to ask is how much the area between the curve representing the function and the X-axis is. This is often referred to as the area under the curve. In this problem, we will approximately calculate the area under the curve for the function \frac{1}{x} between x = 1 and x = e \approx 2.718. We do this approximation by constructing rectangles to cover the area under the curve. We do this in multiple stages, where the number of rectangles we construct in each consecutive stage doubles. As the number of rectangles increases, the top end of the rectangle aligns better with the curve (see the figures below), and the sum of the areas of rectangles approximates the area under the curve better.

    1. Find the sum of areas of rectangles in the following figure.


    2. Find the sum of areas of rectangles in the following figure.


    3. Find the sum of areas of rectangles in the following figure.


    4. Find the sum of areas of rectangles in the following figure.


    5. Can you now estimate the area under the curve 1/x between x = 1 and x = e?

Continuity

Exercises

Problem set

Identify whether each of the following functions is continuous or discontinuous. If discontinuous, identify the points of discontinuity

  1. |x|
  2. \sin x
  3. \frac{1}{x}
  4. \frac{3x^2-5x+1}{2x^2+5x-3}
  5. \cos\left(x + \frac{\pi}{12}\right)
  6. \tan\left(x - \frac{\pi}{2}\right)
  7. e^{2x}
  8. 2\ln 2x-4

Problem set

Identify whether each of the following functions is continuous or discontinuous. If discontinuous, identify the points of discontinuity

  1. f(x) = \begin{cases} 2, & \mbox{if }x = 4 \\ \frac{x-4}{x-4}, & \mbox{if }x \neq 4\end{cases}

  2. g(x) = \begin{cases} 2, & \mbox{if }x = 1 \\ \frac{2x^2-2}{x-1}, & \mbox{if }x\neq 1 \end{cases}

  3. h(x) = \begin{cases} 18, & \mbox{if }x = 3 \\ \frac{3x^2-27}{x-3}, & \mbox{if }x \neq 3\end{cases}

  4. t(x) = \begin{cases} -20, & \mbox{if }x = 2 \\ \frac{20-5x^2}{x-2}, & \mbox{if }x \neq 2\end{cases}

Problem set

Explain with reasons whether each of the following functions is continuous or discontinuous. If discontinuous, identify the points of discontinuity

  1. \sin x + \cos x
  2. \frac{x^2-1}{e^x}
  3. \frac{3^x}{e^x+9}
  4. e^x\cos x-(\ln x)\sin x
  5. \frac{x^2\sin x}{2^x+\sqrt{x}}

Problem set

Explain with reasons whether each of the following functions is continuous or discontinuous. If discontinuous, identify the points of discontinuity

  1. \sin(\cos x)
  2. |16-x^2|-8
  3. e^{x^2}
  4. \sec^{10} x
  5. \sqrt{|\sec x|}
  6. 2+\ln |\cot x|

Problem set

Evaluate the following limits

  1. \lim\limits_{\theta\to \frac{\pi}{2}} \sin \theta

  2. \lim\limits_{\theta\to \frac{\pi}{2}} \csc \theta

  3. \lim\limits_{\theta\to -\frac{\pi}{4}} \tan \theta

  4. \lim\limits_{x\to e} \ln x

  5. \lim\limits_{x\to 0} e^x

Problem set

Evaluate the following limits.

  1. \lim\limits_{x\to 3} (x+1)(x+3)

  2. \lim\limits_{x\to -2} 3x^3-3x^2+3x+1

  3. \lim\limits_{x\to 3} \frac{x^2+1}{x^3+3}

  4. \lim\limits_{x\to 3} \frac{x^2+1}{x^2+3x+3}

  5. \lim\limits_{x\to 1} \frac{x^4-1}{x^2+1}

Problem set

Say, we have the following function definitions:

f(x) = \begin{cases} 2, & \mbox{if }x = 2 \\ \frac{x-2}{x-2}, & \mbox{if }x \neq 2\end{cases}

g(x) = \begin{cases} 4, & \mbox{if }x = 3 \\ \frac{x^2-9}{x-3}, & \mbox{if }x\neq 3 \end{cases}

h(x) = \begin{cases} 20, & \mbox{if }x = 5 \\ 44, & \mbox{if } x=10 \\ 3x+4, & \mbox{otherwise}\end{cases}

Based on the above, evaluate the following.

  1. f(1) and \lim\limits_{x\to 1} f(x)

  2. f(2) and \lim\limits_{x\to 2} f(x)

  3. g(3) and \lim\limits_{x\to 3} g(x)

  4. h(5) and \lim\limits_{x\to 5} h(x)

  5. h(10) and \lim\limits_{x\to 10} h(x)

Problem set

Evaluate the limits in each of the following problems.

Note that, \lceil{x}\rceil is the smallest integer that is greater than or equal to x. For example, \lceil{5.1}\rceil = 6,\,\, \lceil{4.9}\rceil = 5,\,\,\lceil{5}\rceil = 5. And, \lfloor{x}\rfloor is the greatest integer that is less than or equal to x. For example, \lfloor{5.1}\rfloor = 5,\,\, \lfloor{4.9}\rfloor = 4,\,\,\lfloor{5}\rfloor = 5.

  1. \lim\limits_{x\to 1} \lceil{x}\rceil

  2. \lim\limits_{x\to 2.5} \lceil{x}\rceil

  3. \lim\limits_{x\to 3} \lfloor{x}\rfloor

  4. \lim\limits_{x\to 9} \lfloor{x}\rfloor

  5. \lim\limits_{x\to -7.8} \lfloor{x}\rfloor

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to \frac{\pi}{2}} \frac{\sin x \sqrt{\log x}}{e^x}

  2. \lim\limits_{x\to 0} \frac{\sin x \sqrt{\cos x}}{e^x}

  3. \lim\limits_{x\to 0} \log (x^2+1)

  4. \lim\limits_{x\to 0} e^{x^2} - \log (x^2+1)

  5. \lim\limits_{x\to 0} \frac{\cos(\sin x)}{\ln (x+e)}

Limit laws

Exercises

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 3} \frac{x-3}{x-3}

  2. \lim\limits_{x\to 5} \frac{2x-10}{x-5}

  3. \lim\limits_{x\to 3} \frac{x^2-9}{x-3}

  4. \lim\limits_{x\to -2} \frac{x^2-4}{x+2}

  5. \lim\limits_{x\to -7} \frac{49-x^2}{x+7}

  6. \lim\limits_{x\to -3} \frac{x^3+27}{x+3}

  7. \lim\limits_{x\to 2} \frac{x^3-8}{x-2}

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0} \sqrt{\frac{\sin x}{x}}

  2. \lim\limits_{x\to 0} \ln{\frac{\sin x}{x}}

  3. \lim\limits_{x\to 0} \cos\left((-1)^{\left\lfloor 1/x\right\rfloor}x\right)

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0} \left\lfloor x\sin\left(\frac{1}{x}\right)\right\rfloor

  2. \lim\limits_{x\to 0} \left\lfloor x\sin\left(\frac{1}{x}\right)-3\right\rfloor

  3. \lim\limits_{x\to 0} \left\lfloor x\sin\left(\frac{1}{x}\right)+3.5\right\rfloor

  4. \lim\limits_{x\to 0} \left\lceil (-1)^{\left\lfloor 1/x\right\rfloor}x\right\rceil

  5. \lim\limits_{x\to 0} \left\lfloor (-1)^{\left\lfloor 1/x\right\rfloor}x-1\right\rfloor

  6. \lim\limits_{x\to 0} \left\lceil (-1)^{\left\lceil 1/x\right\rceil}x-2.9\right\rceil

Limits involving trigonometric functions

Exercises

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0} \frac{x}{\cos x}

  2. \lim\limits_{x\to 0} \frac{\sin x}{x} + \cos x

  3. \lim\limits_{x\to 0} \frac{10\sin x}{x}

  4. \lim\limits_{x\to 0} \frac{\sin x}{2x}

  5. \lim\limits_{x\to 0} \frac{\sin x}{x(x+3)}

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0} \frac{3\sin x}{xe^x -x^2}

  2. \lim\limits_{x\to 0} \frac{\tan x}{x}

  3. \lim\limits_{x\to 0} \frac{x}{\sin x}

  4. \lim\limits_{x\to 0} x\csc x

  5. \lim\limits_{x\to 0} x\cot x

  6. \lim\limits_{x\to 0} x^2\csc x

  7. \lim\limits_{x\to 0} x\sec x\csc x

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0} \frac{\sin 10x}{x}

  2. \lim\limits_{x\to 0} \frac{\tan (-10x)}{x}

  3. \lim\limits_{x\to 0} x\csc x

  4. \lim\limits_{x\to 0} \frac{\sin x}{\tan x}

  5. \lim\limits_{x\to 0} \frac{\csc x}{\cot x}

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0} \frac{\sin 10x}{\sin (-3x)}

  2. \lim\limits_{x\to 0} \frac{\tan 10x}{\sin (-3x)}

  3. \lim\limits_{x\to 0}\frac{x^2}{\sin x}

  4. \lim\limits_{x\to 0} \frac{\cos\left(\frac{\pi}{2} - x\right)}{\cot\left(\frac{\pi}{2} - x\right)}

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0} \frac{\sin x^2}{x^2}

  2. \lim\limits_{x\to 0} \frac{\tan 2x^3}{x^3}

  3. \lim\limits_{x\to 0} \frac{\tan 2x^2}{\sin x^2}

  4. \lim\limits_{x\to \pi} \frac{\sin x}{x-\pi}

  5. \lim\limits_{x\to \frac{\pi}{2}} \frac{\cos x}{\frac{\pi}{2} - x}

  6. \lim\limits_{x\to \frac{\pi}{2}} \frac{\cot x}{\cos x}

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 1} \frac{\sin (\log x)}{\log x}

  2. \lim\limits_{x\to 1} (\log x)\cot (\log x)

  3. \lim\limits_{x\to 0} \sqrt{x}\csc \sqrt{x}

  4. \lim\limits_{x\to 0} x\csc \sqrt{x}

  5. \lim\limits_{x\to 0}\frac{\sin x^3}{\tan x^2}

Epsilon-delta definition

Exercises

Problem set

  1. If f(x) = 4x, find \delta such that |f(x)-8| < 0.01 whenever |x-2| < \delta.
  2. If g(x) = 2x+5, find \delta such that |g(x)-7| < 0.001 whenever |x-1| < \delta.
  3. If h(a) = x^2, find \delta such that |h(a)-9| < 0.1 whenever |a-3| < \delta.
  4. If t(x) = \frac{\sin x}{x}, use a calculator to find \delta such that |t(x)-1| < 0.01 whenever |x| < \delta.

Problem set

Justify the following using the \varepsilon-\delta definition of a limit.

  1. \lim\limits_{x\to 3}\,\, 2x = 6

  2. \lim\limits_{a\to 1}\,\, -3a+5 = 2

  3. \lim\limits_{p\to 3}\,\, p^2 = 9

  4. \lim\limits_{x\to 2}\,\, 3x^2+4 = 16

  5. \lim\limits_{x\to 9}\,\,\sqrt{x} = 3

  6. \lim\limits_{x\to 3}\,\,-\sqrt{7-x} = -2

  7. \lim\limits_{a\to 2}\,\, 2a^3 = 16

Limits involving infinity

Exercises

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to\infty}\frac{1}{x}

  2. \lim\limits_{x\to\infty}e^{-x}+2

  3. \lim\limits_{x\to\infty}\frac{x^2+2}{3x^2+5}

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to\infty}\frac{-x^{30}-x^{20}-x^{10}}{x^{31}-1}

  2. \lim\limits_{x\to\infty}\frac{10x^3+10x^2+10x+10}{0.1x^4+0.1}

  3. \lim\limits_{x\to\infty}\frac{-2x^5-4x^3-3}{5x^5+4x^4+3x^2+2x+1}

  4. \lim\limits_{x\to\infty}\frac{-6x^3+1}{2x^3+3x^2+4x+1}

  5. \lim\limits_{x\to\infty}\frac{x^4-3x^2+3x-1}{10x^3-9x^2}

  6. \lim\limits_{x\to\infty}\frac{-50x^{50}-25x^{25}+1}{40x^{40}-20x^{20}}

L’Hopital’s rule

For Zero over zero form

Exercises

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0}\frac{\sin 10x}{5x}

  2. \lim\limits_{x\to 1}\frac{\ln x}{x-1}

  3. \lim\limits_{x\to 0}\frac{4\sin 7x}{7\tan 4x}

  4. \lim\limits_{x\to 0}\frac{e^x - 1}{5^x - 1}

  5. \lim\limits_{x\to \frac{\pi}{2}}\frac{1+\cos 2x}{1-\tan x/2}
Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0}\frac{1-\cos x}{x^2}

  2. \lim\limits_{x\to 0}\frac{x^2}{1+\cos x}

  3. \lim\limits_{x\to 0}\frac{\ln(\sec 2x)}{x^2}

  4. \lim\limits_{x\to 0}\frac{x-\sin x}{x\sin x}

  5. \lim\limits_{x\to 0}\frac{\tan x - x}{x\tan x}

  6. \lim\limits_{x\to 0}\frac{3\sin x-3x}{x^3}

  7. \lim\limits_{x\to 0}\frac{x^2 - \sin^2 x}{x^3}

For infinity over infinity form

Exercises

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to \frac{\pi}{2}}\frac{\tan x}{1+3\sec x}
Problem set

Evaluate the following limits

  1. \lim\limits_{x\to \infty}\frac{\ln x}{x}

  2. \lim\limits_{x\to \infty}\frac{(\ln x)^2}{x}

  3. \lim\limits_{x\to \infty}\frac{e^x}{x^{2}}

  4. \lim\limits_{x\to \infty}\frac{\ln x}{\sqrt{x}}

  5. \lim\limits_{x\to \infty}\frac{e^x}{x^{999}}

  6. \lim\limits_{x\to \infty}\frac{\sqrt[99]{x}}{\ln x}
Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0^+}\frac{1}{x\ln x}

  2. \lim\limits_{x\to 0^+}\frac{\ln (\sin x)}{\ln x}

For other indeterminate forms

Exercises

Problem set

Evaluate the following limits

  1. \lim\limits_{x\to 0}{\frac{1}{\tan x} - \frac{1}{x}}

  2. \lim\limits_{x\to \infty}{\ln (x+1)} - {\ln x}
Problem set

Evaluate the following limits

  1. \lim\limits_{x\to \infty}{x^{1/x}}

  2. \lim\limits_{x\to 0^+}x^x

  3. \lim\limits_{x\to \infty}\left(1+\frac{1}{x}\right)^x

  4. \lim\limits_{x\to \infty}\left(1+\frac{3}{x}\right)^x

  5. \lim\limits_{x\to \infty}\left(1+\frac{1}{\sqrt{x}}\right)^x

  6. \lim\limits_{x\to 0^+}\left(1+x\right)^{1/x}

  7. \lim\limits_{x\to 0^-}\left(1+x\right)^{1/x}