Learn rules of continuity, limit of trig and composite function, practice to finding the value of parameter, given the continuity of a piecewise function & continuous in the interval and solve limits with square roots.

Let \('f'\) and \('g'\) be two functions which are continuous at \(x=a\), then the following functions will also be continuous at \(x=a\).

- \(f+g\)
- \(f-g\)
- \(cf\)
- \(fg\)
- \(\dfrac{f}{g}\) if \(g(a)\ne0\)

- We say that :

1. Sum of two continuous function is also continuous.

Proof: Let \(h(x)=f(x)+g(x)\,\forall \,x\)

and at \(x=a\) \(f,\,g\) are say both continuous.

Then, \(\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a^-}f(x)=f(a)\)

and \(\lim\limits_{x\to a^+}g(x)=\lim\limits_{x\to a^-}g(x)=g(a)\)

\(h(a)=f(a)+g(a)\)

\(\lim\limits_{x\to a^+}h(x)=\lim\limits_{x\to a^+}(f(x)+g(x))=\lim\limits_{x\to a^+}f(x)+\lim\limits_{x\to a^+}g(x)\)

\(\lim\limits_{x\to a^-}h(x)=\lim\limits_{x\to a^-}(f(x)+g(x))=\lim\limits_{x\to a^-}f(x)+\lim\limits_{x\to a^-}g(x)\)

\(\therefore\,\,\lim\limits_{x\to a^+}h(x)=\lim\limits_{x\to a^-}h(x)=h(a)\)

\(\therefore\,\,h\) is also continuous at \(x=a\)

Similarly,

2.Difference of two continuous functions is continuous.

3.Product of two continuous functions is continuous.

4.Multiplication by a constant will not change the continuous behavior of a function.

5.Ratio of two continuous functions is continuous. (except where denominator is 0).

- All polynomials are continuous every where i.e in \((–\infty,\,\infty)\).
- All rational functions are continuous whenever they are defined i.e they are continuous in their domain.

- e.g \(f(x)=2x^3+x^2+x-1\) is continuous everywhere

\(f(x)=\dfrac{2x+1}{x-2}\) is continuous for all \(x\) except \(\underbrace{x=2}_{not \,in \,the\,domain}\)

A \(x=–8\)

B \(x=12\)

C \(x=3\)

D \(x=–6\)

All trigonometric functions are continuous at all points where they are defined i.e they are continuous in their domain.

- \(sin\,x,\,cos\,x\) are continuous in R
- \(tan\,x,\,sec\,x\) are continuous in \(R-\left\{(2n+1)\dfrac{\pi}{2}\right\}\)\((n\in I)\)
- \(cosec\,x,\,cot\,x\) are continuous in \(R-\{n\pi\}\) \((n\in I)\)

A \(\dfrac{–1}{5}\)

B 7

C –18

D \(\dfrac{2}{3}\)

The root function \(f(x)= \sqrt[n]{x}=x^{1/n}\) is continuous for all values of \(x\) where it is defined or continuous in its domain.

- Consider the graph of \(f(x)=x^{1/n}\).
- For the case, when \(n\) is even, the graph is similar to \(f(x)=\sqrt x\) as shown in figure.

- For the case, when \(n \) is odd the graph is the graph is similar to \(f(x)=\sqrt[3]{x}\) as shown in the graph.

- In both cases \(f(x)=x^{{1}/{n}}\) is continuous in domain.

If \('g'\) is continuous at \(x=a\) and \('f'\) is continuous at \(x=g(a)\), then the composite function \(f(g(x))\) is continuous at \(x=a\).

- e.g \(h(x)=cos\,x^3\) is continuous at all values of \(x\), because \(h(x)=f(g(x))\) where, \(g(x)=x^3\) and \(f(x)=cos\,x\), both of which are continuous for all \(x\).
- Composite function rule extends for composite of more than two function.

A \(h(x)=sin(2x^2+x+1)\) is continuous at \(x=2\).

B \(h(x)=cos(x^2+x+3)\) is discontinuous at \(x=3\).

C \(h(x)=sin(5x+7)\) is discontinuous at \(x=5\).

D \(h(x)=cos(3x^2+5)\) is discontinuous at \(x=–4\).

\(\lim\limits_{x\to a}f(g(x)) =f\left(\lim\limits_{x\to a }g(x)\right)\)= \(f(b)\). If

(1) \(\lim\limits_{x\to a}\,g(x)=b\)

(2) \('f'\)is continuous at \(x=b\).

e.g

Consider, \(\lim\limits_{x\to a}\,cos(x-2)\)

which is of the form \(\lim\limits_{x\to a}\,f(g(x))\)

where, \(g(x)=x-2\) and \(f(x)=cos\,x\)

\(\therefore\,\lim\limits_{x\to 2}\,cos(x-2)\)

\(=cos\left(\lim\limits_{x\to 2}\,(x-2)\right)\)

\(=cos\,0=1\)

A function \('f'\) is said to be continuous in an interval

[a, b] if it is continuous for all values of \(x\) in [a, b].

- At \(x=a\), continuity means \(\lim\limits_{x\to a^+}\,f(x)=f(a)\) (continuity from right).
- At \(x=b\), continuity means \(\lim\limits_{x\to b^-}\,f(x)=f(b)\) (continuity from left).
- At all the interior points the definition of continuity remain the same i.e R.H.L=L.H.L = \(f(\alpha)\)

Where \(x=\alpha\) is any interior point.

- Even a single point of discontinuity in any interval will make the function discontinuous in the entire interval.

A \(f(x)=\dfrac{sin\,x}{x–8}\)

B \(f(x)=\dfrac{cos\,x}{1–x}\)

C \(f(x)=\dfrac{sin\,x}{x+2}\)

D \(f(x)=\dfrac{cos\,x}{x–4}\)

Suppose \('f'\) is a piecewise defined function.

\(f(x)= \begin{cases} 2x+1 & if & x<2\\ x + 3 & if & x\geq0 \end{cases}\)

then to test continuity at \(f\) at \(x=2\)

R.H.L = \(\lim\limits_{x\to 2^+}f(x)=\lim\limits_{x\to 2^+}(x+3)=5\;\;\;\;\;\;(2^+>2)\)

L.H.L = \(\lim\limits_{x\to 2^-}f(x)=\lim\limits_{x\to 2^-}(2x+1)=5\,\;\;\;\;(2^-<2)\)

\(f(2)=2+3=5\)

\(\therefore\) \(f\) is continuous at \(x=2\)

Now sometimes it is asked after giving continuity ,what is the value of certain parameter.

Consider

\(f(x)= \begin{cases} 2x+\alpha & if & x<2\\ x + 3 & if & x\geq2 \end{cases}\) is continuous at \(x=2\)

then R.H.L = L.H.L = \(f(2)\Rightarrow\,\alpha=1\)

A \(\dfrac{3}{2}\)

B \(\dfrac{5}{11}\)

C –10

D \(\dfrac{–11}{7}\)