L’Hôpital’s rule – bromheads.tv Calculus Tutorial

Consider the limitIf both the numerator and also the denominator room finite in ~ \$a\$ and\$g(a) eq 0\$, climate

Example

\$displaystylelim_x o 3,fracx^2+1x+2=frac105=2\$.

But what wake up if both the numerator and also the denominator have tendency to\$0\$? the is no clear what the limit is. In fact, depending upon whatfunctions \$f(x)\$ and also \$g(x)\$ are, the limit deserve to be anything in ~ all!

Example

\$displaystyleeginarrayl
qquadqquadldisplaystylelim_x o 0, fracx^3x^2=lim_x o 0 x=0. &displaystylelim_x o 0, frac-xx^3=lim_x o 0 frac-1x^2=-infty.\ displaystylelim_x o 0, fracxx^2=lim_x o 0 frac1x=infty. & displaystylelim_x o 0, frackxx=lim_x o 0 k=k.endarray\$

These limits are instances of indeterminate creates of type\$frac00\$. L’Hôpital’s rule provides a methodfor evaluating such limits. We will signify \$displaystylelim_x o a, lim_x o a^+, lim_x o a^-, lim_x o infty, small extrm and also lim_x o -infty\$generically through \$lim\$ in what follows.

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L’Hôpital’s dominion for \$displaystylefrac00\$

Suppose \$lim f(x)=lim g(x)=0\$. Then

If \$displaystyle lim, fracf"(x)g"(x)=L\$, then \$displaystyle lim, fracf(x)g(x)=lim fracf"(x)g"(x)=L\$. If \$displaystyle lim, fracf"(x)g"(x)\$ has tendency to \$+infty\$ or \$-infty\$ in the limit, climate so go \$displaystylefracf(x)g(x)\$.
Geometric Interpretation  Consider \$displaystyle lim_t o a^+, fracy(t)x(t)\$, whereby \$x(a)=y(a)=0\$. In ~ time \$t\$, the secant line v \$(x(t), y(t))\$ and \$(0,0)\$ has slope as \$t o a^+\$, \$x(t) o 0\$ and also \$y(t) o 0\$, and also we expect the secant line to almost right the tangent line at \$(0,0)\$ much better and better. In the limit together \$t o a^+\$, but we can also calculate the slope of the tangent heat at \$(0,0)\$ as

Thus, This is casual geometrical interpretation, and also certainly no a proof, the L’Hôpital’s Rule. However, it does offer us understanding into the official statement of the rule.

Sketch that the proof of L’Hôpital’s preeminence \$displaystyle left(frac00 smallf extrm Case ight)\$

We will use an extension of the typical Value Theorem:

Extended (Cauchy) typical Value Theorem

Let \$f\$ and \$g\$ be differentiable ~ above \$(a,b)\$ and constant on\$\$. Expect that \$g"(x) eq 0\$ in \$(a,b)\$. Climate there is atleast one suggest \$c\$ in \$(a,b)\$ together thatThe proof of this theorem is fairly simple and can be discovered in mostcalculus texts.

We will currently sketch the evidence of L’Hôpital’s dominion for the\$frac00\$ case in the limit together \$x o c^+\$, whereby \$c\$ is finite.The instance \$x o c^-\$ have the right to be proven in a similar manner, and also these twocases together deserve to be offered to prove L’Hôpital’s dominion for atwo-sided limit. This evidence is taken native Salas and Hille’sCalculus: One Variable.

Let \$f\$ and \$g\$ be identified on an interval \$(c,b)\$, whereby \$f(x) o 0\$and \$g(x) o 0\$ together \$x o c^+\$ yet \$displaystyle fracf"(x)g"(x)\$tends to a finite border \$L\$. Climate \$f’\$ and \$g’\$ exist on some set\$(c, c+g>\$ and \$g’ eq 0\$ top top \$(c, c+h>\$. Also, \$f\$ and also \$g\$ arecontinuous on \$\$, wherein we define \$f(c)=0\$ and \$g(c)=0\$.

By the expanded Mean value Theorem, over there exists \$c_hin (c,c+h)\$ suchthat since \$f(c)=g(c)=0\$. Letting \$h o 0^+\$, \$displaystyle lim_h o 0^+,fracf"(c_h)g"(c_h)=lim_x o c^+, fracf"(x)g"(x)\$ while\$displaystyle lim_h o 0^+,fracf(c+h)g(c+h)=lim_x o c^+, fracf(x)g(x)\$. Thus,

Example \$displaystyle lim_x o 0, fracsin xx=lim_x o0, fracfracddx(sin x)fracddx(x)=lim_x o 0,fraccos x1=1.\$ \$displaystyle lim_x o 1, frac2ln xx-1=lim_x o1, fracfracddx(2ln x)fracddx(x-1)=lim_x o 1,frac~frac2x~1=2.\$ \$displaystyle lim_x o 0, frace^x-1x^2=lim_x o0, fracfracddx(e^x-1)fracddx(x^2)=lim_x o 0,frace^x2x= extdoes not exist.\$

If the numerator and also the denominator both often tend to \$infty\$ or\$-infty\$, L’Hôpital’s dominion still applies.

L’Hôpital’s dominance for \$displaystylefracinftyinfty\$

Suppose \$lim f(x)\$ and \$lim g(x)\$ are both infinite. Then

If \$displaystyle lim, fracf"(x)g"(x)=L\$, then\$displaystyle lim, fracf(x)g(x)=lim fracf"(x)g"(x)=L\$. If \$displaystyle lim, fracf"(x)g"(x)\$ tends to \$+infty\$or \$-infty\$ in the limit, climate so go \$displaystylefracf(x)g(x)\$.

The proof of this type of L’Hôpital’s dominion requires much more advancedanalysis.

Here space some examples of indeterminate forms of type\$displaystylefracinftyinfty\$.

Example

\$displaystylelim_x oinftyfrace^xx=lim_x oinfty frace^x1=infty.\$

Sometimes that is essential to usage L’Hôpital’s dominance several time inthe very same problem:

Example

\$displaystylelim_x o 0 frac1-cosxx^2=lim_x o 0fracsin x2x=lim_x o 0fraccos x2=frac12.\$

Occasionally, a limit can be re-written in order come applyL’Hôpital’s Rule:

Example

\$displaystylelim_x o 0, xlnx=lim_x o 0fracln xfrac1x=lim_x o 0,frac~frac1x~-frac1x^2=lim_x o 0, (-x)=0.\$

We have the right to use other tricks to apply L’Hôpital’s Rule. In the nextexample, we usage L’Hôpital’s dominance to evaluate an indeterminate formof type \$0^0\$:

Example

To evaluate \$displaystyle lim_x o 0^+, x^x\$, we will firstevaluate \$displaystyle lim_x o 0^+, ln (x^x)\$.Then due to the fact that \$displaystylelim_x o 0^+, ln (x^x) o 0\$ as \$x o0^+\$ and \$ln (u)=0\$ if and only if \$u=1\$,Thus,

Notice that L’Hôpital’s dominion only uses to indeterminate forms.For the border in the very first example that this tutorial, L’Hôpital’sRule does no apply and also would provide an incorrect an outcome of 6.L’Hôpital’s preeminence is an effective and remarkably easy to usage toevaluate indeterminate forms of type \$frac00\$ and \$fracinftyinfty\$.

Key ConceptsL’Hôpital’s ascendancy for \$frac00\$

Suppose \$lim f(x) = lim g(x) = 0\$. Then

If \$displaystylelim fracf"(x)g"(x) = L,\$ climate \$displaystylelim fracf(x)g(x) = displaystylelim fracf"(x)g"(x) = L\$. If \$displaystylelim fracf"(x)g"(x)\$ often tends to \$+infty\$ or \$-infty\$ in the limit, then so go \$fracf(x)g(x)\$.

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L’Hôpital’s preeminence for \$fracinftyinfty\$ expect \$lim f(x)\$ and \$lim g(x)\$ are both infinite. Then

If \$displaystylelim fracf"(x)g"(x) = L\$, climate \$displaystylelim fracf(x)g(x) = displaystylelim fracf"(x)g"(x) = L\$. If \$displaystylelim fracf"(x)g"(x)\$ often tends to \$+infty\$ or \$-infty\$ in the limit, climate so does \$fracf(x)g(x)\$.