Question

# Find the values of a and b so that the function f given by is continuous at x = 3 and x = 5.

Solution

## Given: We have (LHL at x = 3) = $\underset{x\to {3}^{-}}{\mathrm{lim}}f\left(x\right)=\underset{h\to 0}{\mathrm{lim}}f\left(3-h\right)$$=\underset{h\to 0}{\mathrm{lim}}\left(1\right)=1$                      (RHL at x = 3) = $\underset{x\to {3}^{+}}{\mathrm{lim}}f\left(x\right)=\underset{h\to 0}{\mathrm{lim}}f\left(3+h\right)$$=\underset{h\to 0}{\mathrm{lim}}a\left(3+h\right)+b=3a+b$ (LHL at x = 5) = $\underset{x\to {5}^{-}}{\mathrm{lim}}f\left(x\right)=\underset{h\to 0}{\mathrm{lim}}f\left(5-h\right)$$=\underset{h\to 0}{\mathrm{lim}}\left(a\left(5-h\right)+b\right)=5a+b$                        (RHL at x = 5) = $\underset{x\to {5}^{+}}{\mathrm{lim}}f\left(x\right)=\underset{h\to 0}{\mathrm{lim}}f\left(5+h\right)$$=\underset{h\to 0}{\mathrm{lim}}7=7$ If f(x) is continuous at x = 3 and 5, then  ∴ ​ On solving eqs. (1) and (2), we get MathematicsRD Sharma XII Vol 1 (2015)Standard XII

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