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# If f(x)=[tan(π4+x)]1x,f(x)=[tan(π4+x)]1x,= k,for x=0 is continuous at x=0 , find k. - Mathematics and Statistics

#### Question

If f(x)=[tan(pi/4+x)]^(1/x),

= k                        ,for x=0

is continuous at x=0 , find k.

#### Solution

f(x)=[tan(pi/4+x)]^(1/x), " for "x!=0

f(0)=k

Since f(x) is continuos at x=0

lim_(x->0)f(x)=f(0)

lim_(x->0)[tan(pi/4+x)]^(1/x)=k

lim_(x->0)[(1+tanx)/(1-tanx)]^(1/x)=k

lim_(x->0)[1+(1+tanx)/(1-tanx)-1]^(1/x)=k

lim_(x->0)[1+(1+tanx-1+tanx)/(1-tanx)]^(1/x)=k

lim_(x->0)[1+(2tanx)/(1-tanx)]^(1/x)=k

lim_(x->0)[1+(2tanx)/(1-tanx)]^(1/((2tanx)/(1-tanx))xx((2tanx)/(x.(1-tanx))))=k

e^(lim_(x->0)(2tanx)/(x.(1-tanx)))=k {becauselim_(x->0)[1+x]^(1/x)=e}

e^(2lim_(x->0)(tanx)/(x)xxlim_(x->0)1/(1-tanx))=k {becauselim_(x->0)[tanx/x]=1}

e^(2xx1xx1/(1-0))=k

k=e^2

Is there an error in this question or solution?

#### APPEARS IN

2014-2015 (March) (with solutions)
Question 5.1.3 | 3.00 marks
Solution If f(x)=[tan(π4+x)]1x,f(x)=[tan(π4+x)]1x,= k,for x=0 is continuous at x=0 , find k. Concept: Continuity - Continuity of a Function at a Point.
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