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Pre-Algebra
Mean
Mode
Greatest Common Factor
Least Common Multiple
Order of Operations
Fractions
Mixed Fractions
Prime Factorization
Exponents
Radicals
Algebra
Combine Like Terms
Solve for a Variable
Factor
Expand
Evaluate Fractions
Linear Equations
Quadratic Equations
Inequalities
Systems of Equations
Matrices
Trigonometry
Simplify
Evaluate
Graphs
Solve Equations
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Derivatives
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Solve
Evaluate
\frac{3}{2}=1.5
2
3
=
1
.
5
Quiz
Limits
5 problems similar to:
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}
x
→
−
3
lim
x
2
+
2
x
−
3
x
2
−
9
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Similar Problems from Web Search
What is the value of \displaystyle\lim_{{{x}\rightarrow-{3}}}{\frac{{{x}^{{{2}}}-{9}}}{{{x}^{{{2}}}-{2}{x}-{15}}}} ?
What is the value of
x
→
−
3
lim
x
2
−
2
x
−
1
5
x
2
−
9
?
https://socratic.org/questions/what-is-the-value-of-lim-x-rightarrow-3-frac-x-2-9-x-2-2x-15
\displaystyle\frac{{3}}{{4}} Explanation: \displaystyle\lim_{{{x}\to-{3}}}\frac{{{x}^{{2}}-{9}}}{{{x}^{{2}}-{2}{x}-{15}}} By factoring out the numerator and the denominator, \displaystyle=\lim_{{{x}\to-{3}}}\frac{{\cancel{{{\left({x}+{3}\right)}}}{\left({x}-{3}\right)}}}{{\cancel{{{\left({x}+{3}\right)}}}{\left({x}-{5}\right)}}}=\frac{{-{3}-{3}}}{{-{3}-{5}}}=\frac{{-{6}}}{{-{8}}}=\frac{{3}}{{4}} ...
4
3
Explanation:
x
→
−
3
lim
x
2
−
2
x
−
1
5
x
2
−
9
By factoring out the numerator and the denominator,
=
x
→
−
3
lim
(
x
+
3
)
(
x
−
5
)
(
x
+
3
)
(
x
−
3
)
=
−
3
−
5
−
3
−
3
=
−
8
−
6
=
4
3
...
Evaluating \lim\limits_{x\to-3}\frac{x^2-9}{2x^2+7x+3}
Evaluating
x
→
−
3
lim
2
x
2
+
7
x
+
3
x
2
−
9
https://math.stackexchange.com/q/490053
\require{cancel} Note that both the numerator and denominator evaluate to 0 at x = -3. That means we have an indeterminate limit, which simply means more work needs to be done. Factor ...
Note that both the numerator and denominator evaluate to
0
at
x
=
−
3
. That means we have an indeterminate limit, which simply means more work needs to be done. Factor ...
Help with epsilon-delta proof of \lim_{x \to -3} \frac{x^2 + x - 6}{x^2 - 9}
Help with epsilon-delta proof of
lim
x
→
−
3
x
2
−
9
x
2
+
x
−
6
https://math.stackexchange.com/questions/197844/help-with-epsilon-delta-proof-of-lim-x-to-3-fracx2-x-6x2-9
HINT: \frac{x^2+x-6}{x^2-9}-\frac56=\frac{6x^2+6x-36-(5x^2-45)}{6(x^2-9)}=\frac{x^2+6x+9}{6(x^2-9)}=\frac{(x+3)^2}{6(x-3)(x+3)}
HINT:
x
2
−
9
x
2
+
x
−
6
−
6
5
=
6
(
x
2
−
9
)
6
x
2
+
6
x
−
3
6
−
(
5
x
2
−
4
5
)
=
6
(
x
2
−
9
)
x
2
+
6
x
+
9
=
6
(
x
−
3
)
(
x
+
3
)
(
x
+
3
)
2
What is \displaystyle \lim_{x \to 1} \dfrac{x^2-1}{x^2+3x-4} ?
What is
x
→
1
lim
x
2
+
3
x
−
4
x
2
−
1
?
https://www.quora.com/What-is-displaystyle-lim_-x-to-1-dfrac-x-2-1-x-2+3x-4
You can see if you solve for x=1 that you get a fraction in the form of \dfrac{0}{0} which is indeterminate. When you get into a situation like this you can apply L'Hôpital's rule - Wikipedia. ...
You can see if you solve for
x
=
1
that you get a fraction in the form of
0
0
which is indeterminate. When you get into a situation like this you can apply L'Hôpital's rule - Wikipedia. ...
Proving that \lim_{x\to 2}\frac{x^2-5x}{x^2+2}=-1 using the \epsilon-\delta definition of a limit
Proving that
lim
x
→
2
x
2
+
2
x
2
−
5
x
=
−
1
using the
ϵ
-
δ
definition of a limit
https://math.stackexchange.com/questions/1872290/proving-that-lim-x-to-2-fracx2-5xx22-1-using-the-epsilon-delta
Let \epsilon>0 be given. We have to find a number \delta>0 such that \left|\frac{x^2-5x}{x^2+2}+1\right|<\epsilon whenever |x-2|<\delta. But, as Andre notes, \left|\frac{x^2-5x}{x^2+2}+1\right|=\left|\frac{2x^2-5x+2}{x^2+2}\right|=|x-2|\left|\frac{2x-1}{x^2+1}\right|. ...
Let
ϵ
>
0
be given. We have to find a number
δ
>
0
such that
∣
∣
∣
∣
x
2
+
2
x
2
−
5
x
+
1
∣
∣
∣
∣
<
ϵ
whenever
∣
x
−
2
∣
<
δ
. But, as Andre notes,
∣
∣
∣
∣
x
2
+
2
x
2
−
5
x
+
1
∣
∣
∣
∣
=
∣
∣
∣
∣
x
2
+
2
2
x
2
−
5
x
+
2
∣
∣
∣
∣
=
∣
x
−
2
∣
∣
∣
∣
x
2
+
1
2
x
−
1
∣
∣
∣
.
...
Limit simplification
Limit simplification
https://math.stackexchange.com/q/2704118
Note that \lim_{x\to -3} \frac{(x^2-9)^2}{x^2+\color{red}6x+9} We can solve in this way \lim_{x\to -3} \frac{(x^2-9)^2}{(x+3)^2}=\lim_{x\to -3} \frac{(x-3)^2(x+3)^2}{(x+3)^2}=\lim_{x\to -3} (x-3)^2=36
Note that
lim
x
→
−
3
x
2
+
6
x
+
9
(
x
2
−
9
)
2
We can solve in this way
lim
x
→
−
3
(
x
+
3
)
2
(
x
2
−
9
)
2
=
lim
x
→
−
3
(
x
+
3
)
2
(
x
−
3
)
2
(
x
+
3
)
2
=
lim
x
→
−
3
(
x
−
3
)
2
=
3
6
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\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
[
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]
[
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−
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0
1
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5
]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
{
8
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4
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y
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4
7
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
d
x
d
(
x
−
5
)
(
3
x
2
−
2
)
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
∫
0
1
x
e
−
x
2
d
x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}
x
→
−
3
lim
x
2
+
2
x
−
3
x
2
−
9
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