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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. - Continuity - Continuity of a Function at a Point

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)
Question 5.1.3 | 3 marks
Solution for question: 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. For the courses HSC Arts, HSC Science (Computer Science), HSC Science (Electronics), HSC Science (General)
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