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== Learning Objectives ==
* Having an intuitionistic understanding of Limits of a function.
* Understanding algebra of limits.
* Working out limits of various types of functions.
* Identifying various indeterminate forms.
== Concept 1: Understanding Limits of a function ==
=== Activity ===
===== Understanding the need for limits. =====
=== Theory ===
'''Limits''' : Its surprisingly hard to rigorously define limits but we can deduce an intuitive explanation from various examples. We say that the limit of function <math>f(x)</math> as <math>x</math> tends to ''a'' is ''L'' , if <math>f(x)</math> approaches (gets closer to) ''L'' , as <math>x</math> gets closer to ''a''. If such a number does not exist, we say that the limit does not exist.
This is expressed mathematically as <math>\lim_{x \to a} f(x) = L </math>
=== Activity ===
==== Understanding the definition of limits. ====
=== Important Point ===
What is <math>\lim_{x \to 2} f(x) </math>where <math>f(x) = \left\{
\begin{array}{ll}
x & \quad x \in \R - [2] \\
6 & \quad x = 2
\end{array}
\right.</math>?
Since as <math>x</math> approaches 2, <math>f(x)</math>approaches 2 , <math>\lim_{x \to 2} f(x) </math>= 2.
This shows that <math>\lim_{x \to a} f(x) </math>need not be equal to <math>f(a)</math>even when <math>f(a)</math> is explicitly defined.
== Concept 2 : Limit laws ==
In this section we will learn the algebra of limit of functions.
=== Theory ===
Before delving into more complex theorems, let us establish the basic ones.
Limit of a constant function is same at each point.
i.e. <math> \lim _{x \rightarrow a} c=c
</math>
Also limit of <math> f(x) = x
</math>at <math> a
</math>is <math> a
</math>.
i.e. <math> \lim _{x \rightarrow a} x=a
</math>. This makes sense since the function is defined at all points and is continuous.
Now we are prepared to move forward with the algebra of limits.
<math>\begin{aligned}
&\lim _{x \rightarrow p}(f(x)+g(x))=\lim _{x \rightarrow p} f(x)+\lim _{x \rightarrow p} g(x) \\
&\lim _{x \rightarrow p}(f(x)-g(x))=\lim _{x \rightarrow p} f(x)-\lim _{x \rightarrow p} g(x) \\
&\lim _{x \rightarrow p}(f(x) \cdot g(x))=\lim _{x \rightarrow p} f(x) \cdot \lim _{x \rightarrow p} g(x) \\
&\lim _{x \rightarrow p}(f(x) / g(x))=\lim _{x \rightarrow p} f(x) / \lim _{x \rightarrow p} g(x) \\
&\lim _{x \rightarrow p} \quad f(x)^{g(x)}=\lim _{x \rightarrow p} f(x)^{\lim _{x \rightarrow p} g(x)}
\end{aligned}</math>
=== Activity ===
Intuitively understanding limit laws.
== Concept 3 : Working out limits of functions algebraically ==
Using the limit laws we studied in the last section, we can manipulate functions and try to solve them algebraically instead of using graphs or tables like we did before.
* Having an intuitionistic understanding of Limits of a function.
* Understanding algebra of limits.
* Working out limits of various types of functions.
* Identifying various indeterminate forms.
== Concept 1: Understanding Limits of a function ==
=== Activity ===
===== Understanding the need for limits. =====
=== Theory ===
'''Limits''' : Its surprisingly hard to rigorously define limits but we can deduce an intuitive explanation from various examples. We say that the limit of function <math>f(x)</math> as <math>x</math> tends to ''a'' is ''L'' , if <math>f(x)</math> approaches (gets closer to) ''L'' , as <math>x</math> gets closer to ''a''. If such a number does not exist, we say that the limit does not exist.
This is expressed mathematically as <math>\lim_{x \to a} f(x) = L </math>
=== Activity ===
==== Understanding the definition of limits. ====
=== Important Point ===
What is <math>\lim_{x \to 2} f(x) </math>where <math>f(x) = \left\{
\begin{array}{ll}
x & \quad x \in \R - [2] \\
6 & \quad x = 2
\end{array}
\right.</math>?
Since as <math>x</math> approaches 2, <math>f(x)</math>approaches 2 , <math>\lim_{x \to 2} f(x) </math>= 2.
This shows that <math>\lim_{x \to a} f(x) </math>need not be equal to <math>f(a)</math>even when <math>f(a)</math> is explicitly defined.
== Concept 2 : Limit laws ==
In this section we will learn the algebra of limit of functions.
=== Theory ===
Before delving into more complex theorems, let us establish the basic ones.
Limit of a constant function is same at each point.
i.e. <math> \lim _{x \rightarrow a} c=c
</math>
Also limit of <math> f(x) = x
</math>at <math> a
</math>is <math> a
</math>.
i.e. <math> \lim _{x \rightarrow a} x=a
</math>. This makes sense since the function is defined at all points and is continuous.
Now we are prepared to move forward with the algebra of limits.
<math>\begin{aligned}
&\lim _{x \rightarrow p}(f(x)+g(x))=\lim _{x \rightarrow p} f(x)+\lim _{x \rightarrow p} g(x) \\
&\lim _{x \rightarrow p}(f(x)-g(x))=\lim _{x \rightarrow p} f(x)-\lim _{x \rightarrow p} g(x) \\
&\lim _{x \rightarrow p}(f(x) \cdot g(x))=\lim _{x \rightarrow p} f(x) \cdot \lim _{x \rightarrow p} g(x) \\
&\lim _{x \rightarrow p}(f(x) / g(x))=\lim _{x \rightarrow p} f(x) / \lim _{x \rightarrow p} g(x) \\
&\lim _{x \rightarrow p} \quad f(x)^{g(x)}=\lim _{x \rightarrow p} f(x)^{\lim _{x \rightarrow p} g(x)}
\end{aligned}</math>
=== Activity ===
Intuitively understanding limit laws.
== Concept 3 : Working out limits of functions algebraically ==
Using the limit laws we studied in the last section, we can manipulate functions and try to solve them algebraically instead of using graphs or tables like we did before.