Limits of Continuous Functions
Key Questions

Answer:
the limit is the value of the continuous function at the point concerned
Explanation:
the limit is the value of the continuous function at the point concerned
so for example
#lim_{x to c} x^2 = c^2# 
If the function is continuous at that value of
#x# , just computing it should work. For other functions, try to factor the discontinuity out, and then computing it.If nothing works, and you don't have access to derivatives to use L'HÃ´pital rule, just take a numeric approach and see if it's close to a pattern, or try to use other approaches like the Squeeze Theorem or a geomtrical proof. For example:
#lim_(x rarr4)(x^28x+16) = 0# The function is continuous* for all real values of
#x# so just plugging the value of#x# is good enough.#lim_(xrarr4)(x^216)/(x4) = lim_(xrarr4)((x+4)(x4))/(x4) = lim_(xrarr4)(x+4) = 8# The function isn't continous at
#x=4# because we'd have division by#0# , but using algebra we can take out the denominator, making a continuous function to just plug the number in.#lim_(thetararr0)(sin(theta))/theta = 1# This one you can't factor out. You need to use the unit circle and the Squeeze theorem to make a proof or try a numeric approach.
*A function is continuous on a certain range if you can draw the graph on that range without lifting your writing utensil, or more formally, if there isn't any point that would create a math error, like division by zero, even root of a negative, logarithm of a negative or null number, etc.

Answer:
A continuous function is a function that is continuous at every point in its domain.
That is
#f:A>B# is continuous if#AA a in A, lim_(x>a) f(x) = f(a)# Explanation:
We normally describe a continuous function as one whose graph can be drawn without any jumps. That's a good place to start, but is misleading.
An example of a well behaved continuous function would be
#f(x) = x^3x# graph{x^3x [2.5, 2.5, 1.25, 1.25]}
In fact any polynomial is well defined everywhere and continuous.
A less obvious example of a continuous function is
#f(x) = tan(x)# graph{tan(x) [10, 10, 5, 5]}
This appears to be discontinuous, with 'jumps' at
#x = pi/2+n pi# but those values of#x# are excluded from the domain.Similarly, the following function is continuous on its domain
#(oo, 0) uu (0, oo)# #f(x) = x/abs(x)# graph{x/abs(x) [5, 5, 2.5, 2.5]}
If we add a definition of
#f(0)# then this becomes a discontinuous function.#f(x) = { (x/abs(x), "if x != 0"), (0, "if x = 0") :}# graph{(yx/abs(x))(x^2+y^20.002) = 0 [5, 5, 2.5, 2.5]}
This
#f(x)# fails the condition for continuity at the point#x=0# .#lim_(x>0) f(x)# is not defined, let alone equal to#f(0)#