Question

If f′ is continuous, f(6)=0, and f′(6)=13, evaluate

Answer 1

\[\lim_{x \rightarrow 0} (f(6+3x)+f(6+5x))/x\]

Answer 2

@AllTehMaffs

Answer 3

Gah, haven't done this in forever, and I gotta run - sorry :/

@mashy @shamil98 Can you help?

Answer 4

no worries

Answer 5

@ranga @zepdrix

Answer 6

\[f \prime \left( c \right)=\lim_{x \rightarrow c}\frac{ f \left( x \right)-f \left( c \right) }{ x-c }\]

Answer 7

\[If f \left( x \right) is continuous at any point a then \lim_{x \rightarrow a}f \left( x \right)=f \left( a \right)\]

Answer 8

so how should i answer the question

Answer 9

@muzzammil.raza Have you been taught L'Hopital's Rule yet?

Answer 10

i don't think he is supposed to use L'hospital here :P..

Answer 11

\[ \lim_{x \rightarrow 0} \frac{f(6+3x)+f(6+5x) }{ x } \text{ = } \lim_{x \rightarrow 0} \frac{ f(6+3x)}{ x } \text{ + }\lim_{x \rightarrow 0} \frac{ f(6+5x)}{ x } \]I will do the first limit here: lim x->0 f(6+3x) / x
Write f(6+3x) / x as { f(6+3x) - f(6) } / x {since f(6) = 0}
{ f(6+3x) - f(6) } / x = 3{ f(6+3x) - f(6) } / 3x
Let u = 3x. As x->0, u->0
The limit becomes:
lim u->0 3 { f(6 + u) - f(6) } / u = 3 * lim u->0 { f(6 + u) - f(6) } / u = 3 * f'(6) = 3 * 13 = 39