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  • Examine the continuity of the function f(x) = x^3 + 2x – 1 at x = 1

Examine the continuity of the function

f(x) = x^3 + 2x - 1 at x = 1

Answers (1)

A function f(x) is said to be continuous at x = c if,

Left hand limit (LHL at x = c) = Right hand limit (RHL at x = c) = f(c).
Mathematically we can present it as-

\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) = x^3 + 2x - 1 at x = 1 is continuous   if -

\lim _{h \rightarrow 0} \mathrm{f}(1-\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(1+\mathrm{h})=\mathrm{f}(1)

But we can see that,

\begin{array}{l} {\mathrm{LHL}}=\lim _{h \rightarrow 0} \mathrm{f}(1-\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0}\left\{(1-\mathrm{h})^{3}+2(1-\mathrm{h})-1\right\} \\ \therefore \mathrm{LHL}=(1-0)^{3}+2(1-0)-1=2 \ldots(1) \end{array}
Similarly, we proceed for RHL-
{\mathrm{RHL}}=\lim _{h \rightarrow 0}\left\{(1+h)=\lim _{h \rightarrow 0}\left\{(1+h)^{3}+2(1+h)-1\right\}\right.
\therefore \mathrm{RHL}=(1+0)^{3}+2(1+0)-1=2 \ldots(2)

\\And \\f(1)=(1+0)^{3}+2(1+0)-1=2 \ldots(3) \\ Now from equation 1,2 and 3 we can conclude that \\ \lim _{h \rightarrow 0} \mathrm{f}(1-\mathrm{h})=\lim _{h \rightarrow 0} \mathrm{f}(1+\mathrm{h})=\mathrm{f}(1)=2 \\\therefore f(x) is continuous at x=1

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