Let f(x) be a differentiable function at x = a with f’(a) = 2 and f(a) = 4. Then $\underset{x \to a}{l i m} \frac{x f (a) - a f (x)}{x - a}$ equals:

Option 1 -

2a – 4

Option 2 -

4 – 2a

Option 3 -

a + 4

Option 4 -

2a + 4

2 Views | Posted 2 weeks ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 weeks ago

Correct Option - 2

Detailed Solution:

$\underset{x \to a}{l i m} \frac{x f (a) - a f (x)}{x - a} (\frac{0}{0})$

By L’hospital Rule

$= \underset{x \to a}{l i m} \frac{f (a) - a f' (x)}{1} = f (a) - a f' (a)$

$= 4 - 2 a$

Now equation of line OA be

<!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1819614288"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <munder> <mrow> <mi>l</mi> <mi>i</mi> <mi>m</mi> </mrow> <mrow> <mi>x</mi> <mo>→</mo> <mi>a</mi> </mrow> </munder> <mfrac> <mrow> <mi>x</mi> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>a</mi> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mn>0</mn> </mrow> <mrow> <mn>0</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math> By L’hospital Rule <math> <mrow> <mo>=</mo> <munder> <mrow> <mi>l</mi> <mi>i</mi> <mi>m</mi> </mrow> <mrow> <mi>x</mi> <mo>→</mo> <mi>a</mi> </mrow> </munder> <mfrac> <mrow> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>a</mi> <mi>f</mi> <mo>'</mo> <mrow> <mo> (</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>a</mi> <mi>f</mi> <mo>'</mo> <mrow> <mo> (</mo> <mrow> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>         <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1819614289"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mi>a</mi> </mrow> </math>           <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1819614290"> </o:OLEObject></xml><! [endif]->Now equation of line OA be

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Let f(x) be a differentiable function at x = a with f’(a) = 2 and f(a) = 4. Then $\underset{x \to a}{l i m} \frac{x f (a) - a f (x)}{x - a}$ equals:

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Let f(x) be a differentiable function at x = a with f’(a) = 2 and f(a) = 4. Then l i m x → a x f ( a ) − a f ( x ) x − a equals:

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Let f(x) be a differentiable function at x = a with f’(a) = 2 and f(a) = 4. Then $\underset{x \to a}{l i m} \frac{x f (a) - a f (x)}{x - a}$ equals: