Math

QuestionFind the limit: limh0x+hxh\lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as h approaches0 for the function x+hxh\frac{\sqrt{x+h}-\sqrt{x}}{h}. . We will use the limit properties and algebraic manipulations to simplify the expression.

STEP 2

The given expression is in the form of a difference quotient, which is often difficult to simplify directly. A common technique to simplify expressions of this form is to multiply the numerator and denominator by the conjugate of the numerator.
The conjugate of x+hx\sqrt{x+h}-\sqrt{x} is x+h+x\sqrt{x+h}+\sqrt{x}.
So, we multiply the numerator and denominator by this conjugate.
limh0x+hxh×x+h+xx+h+x\lim{h \rightarrow0} \frac{\sqrt{x+h}-\sqrt{x}}{h} \times \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}

STEP 3

Now, we simplify the numerator using the difference of squares formula (a-b)(a+b) = a^2 - b^2.
limh0(x+h)2(x)2h(x+h+x)\lim{h \rightarrow0} \frac{(\sqrt{x+h})^2 - (\sqrt{x})^2}{h(\sqrt{x+h}+\sqrt{x})}

STEP 4

implify the numerator further.
limh0x+hxh(x+h+x)\lim{h \rightarrow0} \frac{x + h - x}{h(\sqrt{x+h}+\sqrt{x})}

STEP 5

implify the numerator by cancelling the x terms.
limh0hh(x+h+x)\lim{h \rightarrow0} \frac{h}{h(\sqrt{x+h}+\sqrt{x})}

STEP 6

implify the expression by cancelling the h terms.
limh01x+h+x\lim{h \rightarrow0} \frac{1}{\sqrt{x+h}+\sqrt{x}}

STEP 7

Now, we can substitute h =0 into the simplified expression.
limh01x+0+x\lim{h \rightarrow0} \frac{1}{\sqrt{x+0}+\sqrt{x}}

STEP 8

implify the expression further.
limh01x+x\lim{h \rightarrow0} \frac{1}{\sqrt{x}+\sqrt{x}}

STEP 9

implify the expression further.
limh2x\lim{h \rightarrow} \frac{}{2\sqrt{x}}

STEP 10

The limit as h approaches0 is now easy to compute, and it is simply the expression itself, as there are no more h terms.
2x\frac{}{2\sqrt{x}}So, the solution to the problem is 2x\frac{}{2\sqrt{x}}.

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