Note we cannot evaluate this limit directly since the
- denominator is zero when x=A.
Start by factoring the numerator and the - denominator.
- - In this case we find:
-
-
- \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} =
- \lim_{x \to A}\frac{(lin2)(lin1)}{(lin3)(lin1)}
-
-
Since we are taking the limit as x\to A, we may assume that x\ne A.
-
Canceling factors we see:
-
-
- \displaystyle\lim_{x \to A}\frac{(lin2)(lin1)}{(lin3)(lin1)}=
- \lim_{x \to A}\frac{(lin2)}{(lin3)}
-
-
We can evaluate this rational function at x = A.
-
Hence,
-
-
- \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} =
- \frac{D(A) +-B*D}{E(A)+-C*E}.
-
-
Finally,
-
-
- \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} =
- \dfrac{toFraction(D*(A-B)/(E*(A-C)))[0]}{toFraction(D*(A-B)/(E*(A-C)))[1]}.
-
-
\frac{num.text()}{denom.text()}
-
- Evaluate the following limit:
-
-
- \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()}
-
-
Note we cannot evaluate this limit directly since the
denominator is zero when x=A.
\frac{num.text()}{denom.text()}
-
-
- Evaluate the following limit:
-
-
- \displaystyle\lim_{x \to A}\frac{num.text()}{lin1.text()}
-
-
Note we cannot evaluate this limit directly since the
- denominator is zero when x=A.
Start by factoring the numerator.
- - In this case we find:
-
-
- \displaystyle\lim_{x \to A}\frac{num.text()}{lin1} =
- \lim_{x \to A}\frac{(lin2)(lin1)}{lin1}
-
-
Since we are taking the limit as x\to A, we may assume that x\ne A.
-
Canceling factors we see:
-
-
- \displaystyle\lim_{x \to A}\frac{(lin2)(lin1)}{(lin3)(lin1)}=
- \lim_{x \to A}(lin2)
-
-
We can evaluate this linear function at x = A.
-
Hence,
-
-
- \displaystyle\lim_{x \to A}\frac{num.text()}{denom.text()} =
- D(A) +-B*D.
-
-
- Finally, we see the solution is D*(A-B).
-
- Evaluate the following limit:
-
-
- \displaystyle\lim_{x \to A}\frac{x^2-A*A}{x-A}
-
-
Note we cannot evaluate this limit directly since the
- denominator is zero when x=A.
Start by factoring the numerator.
- - In this case we find:
-
-
- \displaystyle\lim_{x \to A}\frac{x^2-A*A}{x-A} =
- \lim_{x \to A}\frac{(x+A)(x-A)}{x-A}
-
-
Since we are taking the limit as x\to A, we may assume that x\ne A.
-
Canceling factors we see:
-
-
- \displaystyle\lim_{x \to A}\frac{(x+A)(x-A)}{x-A} =
- \lim_{x \to A}(x+A)
-
-
We can evaluate this linear function at x = A.
-
Hence,
-
-
- \displaystyle\lim_{x \to A}(x+A) =
- A +A.
-
-
Finally, we see the solution is
-
-
- 2*A.
-
-
- Evaluate the following limit:
-
-
- \displaystyle\lim_{x \to A}\frac{x^3-A*A*A}{x-A}
-
-
Note we cannot evaluate this limit directly since the
- denominator is zero when x=A.
Start by factoring the numerator.
- - In this case we find:
-
-
- \displaystyle\lim_{x \to A}\frac{x^3-A*A*A}{x-A} =
- \lim_{x \to A}\frac{(x^2+Ax +A*A)(x-A)}{x-A}
-
-
Since we are taking the limit as x\to A, we may assume that x\ne A.
-
Canceling factors we see:
-
-
- \displaystyle\lim_{x \to A}\frac{(x^2+Ax +A*A)(x-A)}{x-A} =
- \lim_{x \to A}(x^2+Ax +A*A)
-
-
We can evaluate this quadratic function at x = A.
-
Hence,
-
-
- \displaystyle\lim_{x \to A}(x^2+Ax +A*A) =
- (A)^2 +A(A) + A*A.
-
-
Finally, we see the solution is
-
-
- 3*A*A.
-
-
- Evaluate the following limit:
-
-
- \displaystyle\lim_{x \to A}\frac{x^4-A*A*A*A}{x-A}
-
-
Note we cannot evaluate this limit directly since the
- denominator is zero when x=A.
Start by factoring the numerator.
- - In this case we find:
-
-
- \displaystyle\lim_{x \to A}\frac{x^4-A*A*A*A}{x-A} =
- \lim_{x \to A}\frac{(x^3+Ax^2 +A*Ax + A*A*A)(x-A)}{x-A}
-
-
Since we are taking the limit as x\to A, we may assume that x\ne A.
-
Canceling factors we see:
-
-
- \displaystyle\lim_{x \to A}\frac{(x^3+Ax^2 +A*Ax + A*A*A)(x-A)}{x-A}=
- \lim_{x \to A}(x^3+Ax^2 +A*Ax + A*A*A)
-
-
We can evaluate this cubic function at x = A.
-
Hence,
-
-
- \displaystyle\lim_{x \to A}(x^3+Ax^2 +A*Ax + A*A*A)=
- (A)^3+A(A)^2 +A*A(A) + A*A*A.
-
-
Finally, we see the solution is
-
-
- 4*A*A*A.
-
-
- Evaluate the following limit:
-
-
- \displaystyle\lim_{x \to 0}\frac{\sin(Ax)}{Bx}
-
-
Recall that
-
-
- \displaystyle\lim_{x\to 0}\frac{\sin(x)}{x} = 1.
-
-
Factor out \frac{B}{A} in the denominator to get
-
- \displaystyle\lim_{x \to 0}\frac{\sin(Ax)}{Ax \cdot \frac{B}{A}}
- = \lim_{x \to 0}\frac{\sin(Ax)}{Ax} \cdot \frac{A}{B}
-
-
- Set y = Ax. Since y\to 0 as x\to 0,
-
- \displaystyle\lim_{x \to 0}\frac{\sin(Ax)}{Ax} \cdot \frac{A}{B} = \lim_{y\to 0}
- \frac{\sin(y)}{y} \cdot \frac{A}{B}.
-
-
- Hence
- \displaystyle\lim_{x \to 0}\frac{\sin(Ax)}{Bx}=
- \dfrac{toFraction(A/B)[0]}{toFraction(A/B)[1]}.
-
-
- Evaluate the following limit:
-
-
- \displaystyle\lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A}
-
-
Note we cannot evaluate this limit directly since the
- denominator is zero when x=A*A.
Multiply by the conjugate: -
- -
- \displaystyle\lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} =
- \lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} \cdot\frac{\sqrt{x}+A}{\sqrt{x}+A}
-
-
Now simplify.
-
- \displaystyle\lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} =
- \lim_{x \to A*A}\frac{(x-A*A)(\sqrt{x}+A)}{x-A*A}
-
-
Since we are taking the limit as x\to A*A, we may assume that x\ne A*A.
-
Canceling factors we see:
-
- \displaystyle\lim_{x \to A*A}\frac{x-A*A}{\sqrt{x}-A} =
- \lim_{x \to A*A}(\sqrt{x}+A)
-
-
We can evaluate this function at x = A*A.
-
Hence, we see the solution is 2*A
-