Limit of a functionYou read x on the X-axis (horizontal), if ("x tends to infinity"), it means that you've got to go far away towards the right on the X-axis. On the contrary the f(x) values are read on the vertical Y-axis. Hence if it means that when x approaches infinity, the corresponding f(x) values approach to infinity too (go up on the vertical axis). To understand well, observe the 4 following situations below.
asymptotsDon't be afraid by this word ! An asympote is a line that gets closer and closer to the curve but never intersects it. There are 3 kind of asymptots, the horizontal, vertical and slant ones.
Main limitsHere are the limits of the more common functions :
For the inverse function, there are four limits because when x tends to 0 you have to distinguish two cases, if x tends 0 but stays on the right (thats is x is positive) then the limit is , if x tends to 0 from the left then the limit is .
Calculation of limitsThe limit of a sum of functions is the sum of the limits of the functions and the limit of a product of functions is the product of the limits of the functions, the same for division and subtraction. However be careful with infinity or divisions by 0.
You can't calculate the limit :
- of a difference of functions if both tends to +infinity.
- of a product of functions if one tends to 0 and the other to infinity.
- of a quotient of functions if both tend to 0 or infinity.
We call them indeterminate forms. For all the other cases, it's logical : infinity added to a number is infinity, infinity multiplied by a number is infinity, zero divided by infinity is 0, a number divided by infinity is zero, etc...
When you face an indeterminate form the common solution consists in factorizing by the term of greatest degree. Here are two example of calculation of limits.
Examples*** Determine .
This is an indeterminate form because the first term tends to infinity, the second too and we can't deduce anything from infinity minus infinity. Hence we factorize
We obtain a product, the limit of x cubed is known but we must do a more detailed calculation for the second factor :
This is also an indeterminate form because the numerator tends to and the denominator tends to - thus we factorize both the numerator and the denominator then we cancel the result.
Write the calculation then conclude.