Introduction to Limits

Key Questions

• What is the purpose of a limit in calculus?

  A limit allows us to examine the tendency of a function around a given point even when the function is not defined at the point. Let us look at the function below.
  #f(x)={x^2-1}/{x-1}#
  Since its denominator is zero when #x=1#, #f(1)# is undefined; however, its limit at #x=1# exists and indicates that the function value approaches #2# there.
  #lim_{x to 1}{x^2-1}/{x-1} =lim_{x to 1}{(x+1)(x-1)}/{x-1} =lim_{x to 1}(x+1)=2#

  This tool is very useful in calculus when the slope of a tangent line is approximated by the slopes of secant lines with nearing intersection points, which motivates the definition of the derivative.

  Wataru · 1 · Sep 6 2014
• What exactly is a limit in calculus?

  A limit allows us to examine the tendency of a function around a given point even when the function is not defined at the point. Let us look at the function below.
  #f(x)={x^2-1}/{x-1}#
  Since its denominator is zero when #x=1#, #f(1)# is undefined; however, its limit at #x=1# exists and indicates that the function value approaches #2# there.
  #lim_{x to 1}{x^2-1}/{x-1} =lim_{x to 1}{(x+1)(x-1)}/{x-1} =lim_{x to 1}(x+1)=2#

  This tool is very useful in calculus when the slope of a tangent line is approximated by the slopes of secant lines with nearing intersection points, which motivates the definition of the derivative.

  Wataru · 1 · Sep 8 2014

• How doI find limits in calculus?
• How do limits work in calculus?
• What exactly is a limit in calculus?
• What is the purpose of a limit in calculus?
• What is rational function and how do you find domain, vertical and horizontal asymptotes. Also what is "holes" with all limits and continuity and discontinuity?
• lim x-->-1- f(x) = ?
• How do you use the Squeeze Theorem to show that #limsinx/x# as x approaches infinity?
• How do you use the Squeeze Theorem to show that #sqrt (x) * e^(sin(pi/x))=0# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim xcos(1/x)# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim x^2 (Sin 1/x)^2 # as x approaches zero?
• How do you use the Squeeze Theorem to find #lim (arctan(x) )/ (x)# as x approaches infinity?
• How do you use the Squeeze Theorem to find #lim Tan(4x)/x# as x approaches infinity?
• How do you use the Squeeze Theorem to find #lim (sin^2x/x^2)# as x approaches infinity?
• How do you use the Squeeze Theorem to find #lim ( sin (2pi/x))# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim ((sin n )^2) / 2^n# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim (1/x)cosx# as x approaches infinity?
• How do you use the Squeeze Theorem to find #lim sqrt(x)sin(1/x) # as x approaches infinity?
• How do you use the Squeeze Theorem to find #lim sqrtx[1+ sin^2 (2π /x)]# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim tan(x)cos(sin(1/x))# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim(x-1)sin(pi/x-1) # as x approaches one?
• How do you use the Squeeze Theorem to find #limsin(1/x)# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim xcos(50pi/x)# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim sqrt(x^3+x^2) * sin (pi/x) # as x approaches zero?
• How do you use the Squeeze Theorem to find #lim (x^2)(cos20(pi*x)) # as x approaches zero?
• How do you use the Squeeze Theorem to find #lim (1-cos(x))/x# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim Sin(x)/x# as x approaches zero?
• How do you use the Squeeze Theorem to find #lim Arctan(n^2)/sqrt(n)# as x approaches infinity?
• lim x--> ?
• How do you prove the statement lim as x approaches 2 for #(x^2 - 4x + 5) = 1# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 0 for #x^2 = 0# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 4 for # (x^2) = 4# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches -3 for #(x^2+3x)# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 6 for #((x/4)+3) = 9/2# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 2 for #((x^2+x-6)/(x-2))=5# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches -1.5 for # ((9-4x^2)/(3+2x))=6# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 1 for #[ x + 1 ] / [ 2x + 1 ] = 2/3# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 0 for #x^3=0# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 4 for #(7 – 3x) = -5# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 3 for #(x^2+x-4) = 8# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 2 for # (x^2 - 3x) = -2# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 3 for #(x/5) = 3/5# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 1 for #(5x^2)=5# using the epsilon and delta definition?
• How do you prove the statement lim as x approaches 9 for #root4(9-x) = 0# using the epsilon and delta definition?
• lim h -> 0 ?
• How to sketch a graph of this function and determine if f is continuous at x = 3?
• Question #8cc23
• What simple rigorous ways are there to incorporate infinitesimals into the number system and are they then useful for basic Calculus?
• Consider the function #f(x)= -(x-3)^2+4# how do you write an equation using a limit to determine the area enclosed by f(x) and the x-axis?
• Question #17f7b
• Question #969c8
• Question #fcdc0
• How do I teach myself calculus?
• Question #24348
• Anyone can explain to me what's the difference between "limit", "limsup" and "liminf" of a function? It would be helpful to explain with concrete example.
• #lim_(x->oo)((x - 1)/(x + 1) )^x =# ?
• #P in NN#\{0,1};#q>0#;how to solve limit?#lim_(n->oo)(qn+1)/(qn)*(qn+p+1)/(qn+p)*...*(qn+np+1)/(qn+np)#
• When do we know that a limit of a given function exists?
• How did earlier mathematicians calculate limits so accurately?
• What is # lim_(x rarr 0) (h(3+x)-h(3) )#?
• If #f(x)=lim_(n->oo)1/(1+nsin^2pix)# , find the value of #f(x)# for all real values of #x#?
• Question #62181
• Question #d5a95
• Evaluate: #lim_(x→0) (arcsinx-x)/x^3#?
• #lim_(xrarrpi/2) 1-sinx/(x-π/2)#?