Calculus Examples

$lim_{x \to \infty} \sqrt{49 x^{2} + x} - 7 x$

Step 1

Change the two-sided limit into a right sided limit.

$lim_{x \to \infty^{+}} \sqrt{49 x^{2} + x} - 7 x$

Step 2

Multiply to rationalize the numerator.

$lim_{x \to \infty^{+}} \frac{(\sqrt{49 x^{2} + x} - 7 x) (\sqrt{49 x^{2} + x} + 7 x)}{\sqrt{49 x^{2} + x} + 7 x}$

Step 3

Simplify.

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Step 3.1

Expand the numerator using the FOIL method.

$lim_{x \to \infty^{+}} \frac{{\sqrt{49 x^{2} + x}}^{2} + \sqrt{49 x^{2} + x} (7 x) + \sqrt{49 x^{2} + x} (- 7 x) - 49 x^{2}}{\sqrt{49 x^{2} + x} + 7 x}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $49 x^{2}$ from $49 x^{2}$ .

$lim_{x \to \infty^{+}} \frac{x + 0}{\sqrt{49 x^{2} + x} + 7 x}$

Step 3.2.2

Add $x$ and $0$ .

$lim_{x \to \infty^{+}} \frac{x}{\sqrt{49 x^{2} + x} + 7 x}$

Step 4

Factor $x$ out of $49 x^{2} + x$ .

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Step 4.1

Factor $x$ out of $49 x^{2}$ .

$lim_{x \to \infty^{+}} \frac{x}{\sqrt{x (49 x) + x} + 7 x}$

Step 4.2

Raise $x$ to the power of $1$ .

$lim_{x \to \infty^{+}} \frac{x}{\sqrt{x (49 x) + x^{1}} + 7 x}$

Step 4.3

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty^{+}} \frac{x}{\sqrt{x (49 x) + x \cdot 1} + 7 x}$

Step 4.4

Factor $x$ out of $x (49 x) + x \cdot 1$ .

$lim_{x \to \infty^{+}} \frac{x}{\sqrt{x (49 x + 1)} + 7 x}$

Step 5

Because there are no values to the right of $\infty$ in the domain of $\frac{x}{\sqrt{x (49 x + 1)} + 7 x}$ , the limit does not exist.

$Does not exist$