A function gives the outputs for various inputs of the variable. The limit of the function as the variable approaches a particular value of the variable is nothing but the value of the function at that point. Sometimes, due to the its nature, the function may not be defined for some values of the variable. But it may be possible to simplify the given function and the limit may be calculated by imposing a restriction on the variable. Although, the function may not exist at that point, the limit may exist. There is however a condition for a limit to exist at a particular point. The limits as calculated when the variable approaches a point from both sides of the point must be the same. The concept of limits is very important in finding the asymptotes of a function.
The limit of a function f(x) when the variable approaches to the value ‘a’ is symbolically denoted as,
Definition of a Limit
A function varies according to the variation of the independent variable. Let us consider a point x = c on the function. The variable may approach the point c from both the directions. That is either from the positive side of c (as per number line concept from the right side) or from the negative side of c (left side). The limit of the function at c is defined the value of the function at x = c, if the function is defined there, or the value it is supposed to reach if the function is not defined there.
The concept of the limit of a function can be better understood with a real life example. Suppose, you are walking along a road and the road ends in a park. The limit of your ultimate reach is actually the park as you can be in the park. Instead, if the road ends in a well, here again the limit of your reach is up to the closest point near the well but not exactly the well. Obviously your existence in the well is not defined but despite that you can say that your limit is the well.
The concept of the limit of a function can be better understood with a real life example. Suppose, you are walking along a road and the road ends in a park. The limit of your ultimate reach is actually the park as you can be in the park. Instead, if the road ends in a well, here again the limit of your reach is up to the closest point near the well but not exactly the well. Obviously your existence in the well is not defined but despite that you can say that your limit is the well.
How to Solve Limits
The limit of composite functions follows simple laws of operations,
$\lim_{x \to a}$ (a) f(x) = (a) $\lim_{x \to a}$ f(x) , where 'a' is a constant.
$\lim_{x \to a}$ [f(x) $\pm$ g(x)] = $\lim_{x \to a}$ f(x) $\pm$ $\lim_{x \to a}$ g(x)
$\lim_{x \to a}$ [f(x) * g(x)] = $\lim_{x \to a}$ f(x) * $\lim_{x \to a}$ g(x)
$\lim_{x \to a}$ [f(x) $\div$ g(x)] = $\lim_{x \to a}$ f(x) $\div$ $\lim_{x \to a}$ g(x)
As mentioned earlier the limit of a function is the value of the function at that point, if the function is defined there.
That is, if f(a) exists,
If a function f(x) is not defined at a point 'a' but if f(x) is simplified as g(x) with an assumption x $\rightarrow$ a, then the limit can be evaluated as g(a), if it exists.
Sometimes, the direct limit of a rational function f(x) $\div$ g(x) takes the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, such forms are called indeterminate forms, as per L'Hospital rule, the limit of the given rational function is same as $\lim_{x \to a}$ [f '(x) $\div$ g '(x)] and if it still leads to indeterminate form the process may be continued. However it may clearly be noted that this rule is applicable only when the limit leads to an indeterminate form.
Limit Laws
We have a number of laws on operation with limits of different functions that help us to evaluate the over all limit.
The limit of the sum or the difference of two functions is the sum or difference of limits of individual functions, the signs are in the same order. That is,
$\lim_{x\rightarrow a}[f(x)\pm g(x)]=\lim_{x\rightarrow a}[f(x)]\pm \lim_{x\rightarrow a}[g(x)]$
The limit of the product of two functions is product of limits of individual functions. That is,
$\lim_{x\rightarrow a}[f(x)\ast g(x)]=\lim_{x\rightarrow a}[f(x)]\ast \lim_{x\rightarrow a}[g(x)]$
The limit of the quotient of two functions is the quotient of limits of individual functions, the limit of the quotient function not equal to 0. That is,
$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}[f(x)]}{\lim_{x\rightarrow a}[g(x)]},\lim_{x\rightarrow a}[g(x)]\neq 0$
The limit of a function raised to a power is the limits of the function, raised to the same power. That is,
$\lim_{x\rightarrow a}[f(x)]^{n}=\left [ \lim_{x\rightarrow a}f(x) \right ]^{n}$
The limit of a composite function is function of the limit of the inner function. That is,
$\lim_{x\rightarrow a}f[g(x)]=f[\lim_{x\rightarrow a}g(x)]$
In addition, the following law is very important and it is called the squeeze law.
If f(x) ≤ g(x) ≤ h(x), and if, a is a point in the interval covered by all the functions,
$\lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a}h(x)=l$
then,
$\lim_{x\rightarrow a}g(x)=l$
The limit of the sum or the difference of two functions is the sum or difference of limits of individual functions, the signs are in the same order. That is,
$\lim_{x\rightarrow a}[f(x)\pm g(x)]=\lim_{x\rightarrow a}[f(x)]\pm \lim_{x\rightarrow a}[g(x)]$
The limit of the product of two functions is product of limits of individual functions. That is,
$\lim_{x\rightarrow a}[f(x)\ast g(x)]=\lim_{x\rightarrow a}[f(x)]\ast \lim_{x\rightarrow a}[g(x)]$
The limit of the quotient of two functions is the quotient of limits of individual functions, the limit of the quotient function not equal to 0. That is,
$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}[f(x)]}{\lim_{x\rightarrow a}[g(x)]},\lim_{x\rightarrow a}[g(x)]\neq 0$
The limit of a function raised to a power is the limits of the function, raised to the same power. That is,
$\lim_{x\rightarrow a}[f(x)]^{n}=\left [ \lim_{x\rightarrow a}f(x) \right ]^{n}$
The limit of a composite function is function of the limit of the inner function. That is,
$\lim_{x\rightarrow a}f[g(x)]=f[\lim_{x\rightarrow a}g(x)]$
In addition, the following law is very important and it is called the squeeze law.
If f(x) ≤ g(x) ≤ h(x), and if, a is a point in the interval covered by all the functions,
$\lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a}h(x)=l$
then,
$\lim_{x\rightarrow a}g(x)=l$
Upper and Lower Limits
In the earlier section we mentioned that a variable can approach any point in the domain from either the right side or the left side. The limit of the function when approached from the right side of the point is called as the upper limit of the function at that point. Similarly, the limit of the function when approached from the left side of the point is called as the lower limit of the function at that point. These limits are indicated as follows. Upper limit Lower limit
$\lim_{x\rightarrow c^{+}}f(x)$ $\lim_{x\rightarrow c^{-}}f(x)$The over all limit of the function at a given point is said to exist if and only if both the upper limit and the lower limit are equal. The over all limit is denoted as below
$\lim_{x\rightarrow c}f(x)$
Let us illustrate the above concept with a few examples:
$\lim_{x\rightarrow c^{+}}f(x)$ $\lim_{x\rightarrow c^{-}}f(x)$The over all limit of the function at a given point is said to exist if and only if both the upper limit and the lower limit are equal. The over all limit is denoted as below
$\lim_{x\rightarrow c}f(x)$
Let us illustrate the above concept with a few examples:
Solved Examples
Question 1: Let f(x) = 2x + 5. Find the limit as x -> 2
Solution:
Solution:
The function is defined at x = 2 and hence the limit of the function at x = 2, whether the variable
approaches from positive side or negative side, is f(2) = 2(2) + 5 = 9. That is, the upper limit is 9 and
the lower limit also is 9. Since both limits are equal, the limit of the function at x = 2 exists and it is
equal to 9.
approaches from positive side or negative side, is f(2) = 2(2) + 5 = 9. That is, the upper limit is 9 and
the lower limit also is 9. Since both limits are equal, the limit of the function at x = 2 exists and it is
equal to 9.
Question 2: Let f(x) = $\frac{x^{2}-9}{x-3}$. Find the limit as x -> 3
Solution:
Clearly the function is not defined at x = 3. But with an assumption that x $neq$ 3, the function can be
simplified as (x + 3) and hence the limit of the function at x = 3, whether the variable
approaches from positive side or negative side, is f(3) = 3 + 3 = 6. That is, the upper limit is 6 and
the lower limit also is 6. Since both limits are equal, the limit of the function at x = 3 exists and it is
equal to 9 despite the fact that the function is not defined at x = 3.
Solution:
Clearly the function is not defined at x = 3. But with an assumption that x $neq$ 3, the function can be
simplified as (x + 3) and hence the limit of the function at x = 3, whether the variable
approaches from positive side or negative side, is f(3) = 3 + 3 = 6. That is, the upper limit is 6 and
the lower limit also is 6. Since both limits are equal, the limit of the function at x = 3 exists and it is
equal to 9 despite the fact that the function is not defined at x = 3.
Question 3: Let f(x) = 2x + 5, when x < 0, and
= 2x – 5, when x ≥ 0. Find the limit of the function at x = 0.
Solution:
The function is defined at x = 0 because as per the interval of the piecewise function f(0) = -5, which
is also the upper limit of the function at x = 0. However when x approaches 0 from negative side it only
tends to reach 2(0) + 5 = 5. In other words, the lower limit of the function at x = 0 is 5.
But since upper limit and the lower limit are not same, the limit of the function at x = 0 does not exist.
= 2x – 5, when x ≥ 0. Find the limit of the function at x = 0.
Solution:
The function is defined at x = 0 because as per the interval of the piecewise function f(0) = -5, which
is also the upper limit of the function at x = 0. However when x approaches 0 from negative side it only
tends to reach 2(0) + 5 = 5. In other words, the lower limit of the function at x = 0 is 5.
But since upper limit and the lower limit are not same, the limit of the function at x = 0 does not exist.
In the previous section we discussed upper limit and lower limit of a function at a given point. These are also called as the right hand side limit and the left hand side limit respectively. Although the limit at a given point does not exist if these two limits are not the same, sometimes, our study may be restricted to the limit when the variable approaches from any one of the sides. Such limits are called as one-sided limits.
But when we mention about any one-sided limit, it is important that the direction must be specified. In example 3 in the previous section, the right hand side limit at x = 0 is -5 and the left hand side limit at x = 0 is 5. Similarly if f(x) = $\frac{1}{x}$, the right hand side limit at x = 0 is $\infty$ and the left hand side limit at x = 0 is -$\infty$
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But when we mention about any one-sided limit, it is important that the direction must be specified. In example 3 in the previous section, the right hand side limit at x = 0 is -5 and the left hand side limit at x = 0 is 5. Similarly if f(x) = $\frac{1}{x}$, the right hand side limit at x = 0 is $\infty$ and the left hand side limit at x = 0 is -$\infty$
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When the variable of a function approaches
$\infty$ or -$\infty$ , the value of the function depends on the type of function and may not be predictable.
But some rational functions exhibit interesting results when the variable approaches $\infty$ or -$\infty$. If the degree of the numerator function is less than that of the denominator function then the limit at infinity of the function is 0. For example, if $f(x)=[\frac{5x}{(2x^{2}+3x)}]$, dividing both the numerator and the denominator by x, the numerator is free from the variable and the variable will only be at the denominator part. Hence the limit at infinity of this function is 0.
Similarly if the degree of the numerator function is the same as that of the denominator function, the limit at the infinity of the function is the ratio of the leading coefficients. For example, if $f(x)=[\frac{5x^{2}}{(2x^{2}+3x)}]$, dividing both the numerator and the denominator by x2, the leading terms of both the numerator and the denominator is free from the variable. The remaining terms vanish when the variable tends to $\infty$ or -$\infty$ . Hence the limit at infinity of this function is $\frac{5}{2}$.
The straight line equation y = value of the limit at infinity of a function is called the horizontal asymptote of the function.
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But some rational functions exhibit interesting results when the variable approaches $\infty$ or -$\infty$. If the degree of the numerator function is less than that of the denominator function then the limit at infinity of the function is 0. For example, if $f(x)=[\frac{5x}{(2x^{2}+3x)}]$, dividing both the numerator and the denominator by x, the numerator is free from the variable and the variable will only be at the denominator part. Hence the limit at infinity of this function is 0.
Similarly if the degree of the numerator function is the same as that of the denominator function, the limit at the infinity of the function is the ratio of the leading coefficients. For example, if $f(x)=[\frac{5x^{2}}{(2x^{2}+3x)}]$, dividing both the numerator and the denominator by x2, the leading terms of both the numerator and the denominator is free from the variable. The remaining terms vanish when the variable tends to $\infty$ or -$\infty$ . Hence the limit at infinity of this function is $\frac{5}{2}$.
The straight line equation y = value of the limit at infinity of a function is called the horizontal asymptote of the function.
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We have discussed about the limits of the functions when the variable approaches infinity. Now, let us study when the function tends
$\infty$ or -$\infty$ .
For example, if f(x) = $\frac{1}{x}$, the function tends to $\infty$ when the variable approaches 0 from positive side but tends to -$\infty$ when the variable approaches from negative side.
If the function is of the form $\frac{1}{[(x+a)(x-b)]}$, the function tends to infinity at two points. That is when x = -a and when x = -b.
In general any rational function tends to $\infty$ or -$\infty$ , when the denominator part becomes 0. Let us say a rational function tends to $\infty$ or -$\infty$ at x = c. Then the graph of the function will tend to meet the vertical line of equation x = c, but it never meets actually. Such vertical lines are called as vertical asymptotes of the function.
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For example, if f(x) = $\frac{1}{x}$, the function tends to $\infty$ when the variable approaches 0 from positive side but tends to -$\infty$ when the variable approaches from negative side.
If the function is of the form $\frac{1}{[(x+a)(x-b)]}$, the function tends to infinity at two points. That is when x = -a and when x = -b.
In general any rational function tends to $\infty$ or -$\infty$ , when the denominator part becomes 0. Let us say a rational function tends to $\infty$ or -$\infty$ at x = c. Then the graph of the function will tend to meet the vertical line of equation x = c, but it never meets actually. Such vertical lines are called as vertical asymptotes of the function.
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Rates of Change and Limits
Consider a function f(x) continuous in a closed interval [a, b]. The change in the function over this interval is f(b) - f(a) when the variable changes from the value a to b. In other words, the change in value of the variable is (b - a).
The ratio of the change in the function for a change in the variable of in this interval is given by, $\frac{[f(b)-f(a)]}{[b-a]}$. This is called as the average rate of change of the function in the interval [a, b].
Now let us consider a is any point x in the interval and b be infinitesimally close to a by an unit h. That is, a = x and b = x + h and hence b - a = h. Therefore, the average rate of change under this condition at the neighborhood of any point x is, $\frac{[f(x+h)-f(x)]}{[h]}$.
This ratio is called as the difference quotient of the function at any point x. The limit of this quotient as h -> 0, is the instantaneous rate of change of the function at that point , known as the derivative of the function at that point. Graphically it is the slope of the tangent to the graph of the function at that point.
The ratio of the change in the function for a change in the variable of in this interval is given by, $\frac{[f(b)-f(a)]}{[b-a]}$. This is called as the average rate of change of the function in the interval [a, b].
Now let us consider a is any point x in the interval and b be infinitesimally close to a by an unit h. That is, a = x and b = x + h and hence b - a = h. Therefore, the average rate of change under this condition at the neighborhood of any point x is, $\frac{[f(x+h)-f(x)]}{[h]}$.
This ratio is called as the difference quotient of the function at any point x. The limit of this quotient as h -> 0, is the instantaneous rate of change of the function at that point , known as the derivative of the function at that point. Graphically it is the slope of the tangent to the graph of the function at that point.